1
In standard form, c for x + y = 7 equals —
💡 Hint:
Move 7 to the left.
Explanation:
x + y − 7 = 0, so c = −7.
Class 10 · Maths · Chapter 3 · बिहार बोर्ड (BSEB)CBSE · NCERT 2026-27
Pair of Linear Equations in Two Variables
Pair of Linear Equations in Two Variables
How to use this page:
1. Read — Graph · substitution · elimination · ratio test · cross-multiplication in the older book, diagram, worked example, board tip
2. Check — each lesson has its own questions; the number follows the lesson
3. Mastery ★ — all of that lesson correct. Redo the wrong ones
4. The memory figure shows three pairs of lines — crossing lines with one solution, parallel lines with none, and coincident lines with infinitely many.
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- 1 The pair and the standard form
- 2 The graphical method — three pictures
- 3 The substitution method
- 4 The elimination method
- 5 Consistent or inconsistent from the ratios
- 6 Cross-multiplication — the older Bihar book
- Chapter winner — every lesson at mastery ★
🎬 Shorts
This chapter has no Short on the channel yet, so ten empty cards are not shown. The notes, diagrams and checks are complete.
📄 PDF download — notes + answer key
The pair and the standard form
युग्म और मानक रूप · NCERT 3.1 · a1x + b1y + c1 = 0
Two equations, taken togetherNotes
A linear equation in two variables is ax + by + c = 0. a and b are not both zero. When two such equations apply together to the same x and y, that is a pair of linear equations.
A solution is a pair (x, y) that makes both equations true at once. One equation alone gives a whole line. The pair picks the point on that line which also lies on the second line. The sign of the constant flips when it moves to the left.
Turn one situation into two equationsActivity
A shop sold x pens and y notebooks. In all, 40 items were sold, and there were 8 more pens than notebooks. Write x + y = 40 and x = y + 8. The second can also be written x − y − 8 = 0. Together they are a pair. The first line alone is not enough, because many pairs sit on it.
Worked exampleExample
Question: Write 3x + 2y = 12 as a1x + b1y + c1 = 0 and name a1, b1, c1.
Formula: a1x + b1y + c1 = 0.
Substitution: 3x + 2y − 12 = 0.
Answer: a1 = 3, b1 = 2, c1 = −12. These are exact values. Do not write c1 as +12.
10-second revision
- The standard form is ax + by + c = 0
- A solution fits both equations at once
- The constant changes sign when it moves left
Board tip · BSEBBoard tip
In a BSEB answer write all three of a1, b1 and c1. Leaving only x + y = 5 keeps the standard form incomplete.
Board tip · CBSEBoard tip
On CBSE the sign error is common. For x + y = 5, c = −5, not +5.
Check your understandingall correct = mastery ★
2
A solution of the pair makes both equations true at once.
Explanation:
True — sitting on only one equation is not enough.
3
In ax + by + c = 0, a and b are not both ______.
💡 Hint:
Otherwise it is not a linear equation.
Explanation:
Zero.
4
x + y = 40 and x = y + 8 together are —
💡 Hint:
Two equations, the same two variables.
Explanation:
This is a linear pair in two variables.
5
Write 2x − y = 4 in standard form and name a, b, c.
Model answer (board format)
Formula: ax + by + c = 0. Substitution: 2x − y − 4 = 0. Exact values a = 2, b = −1, c = −4.
Check your answer:
The graphical method — three pictures
आलेख विधि — तीन तस्वीरें · NCERT 3.2 · Exercise 3.1 (current book)
Each equation is a straight lineNotes
The graph of each linear equation is a straight line. The two lines of a pair can sit in three ways. If they cross at one point there is one solution, if they are parallel there is none, and if they coincide there are infinitely many.
A pair with at least one solution is consistent. A pair with no solution is inconsistent. Coincident lines are called a dependent pair, and a dependent pair is always consistent.
Draw each line through two pointsActivity
Exercise 3.1 of the current book is this job. Take two easy points on each equation, such as x = 0 and y = 0. Use one scale on both axes. Put the meeting point back into both equations and check. In the older Bihar book this graph is exercise 3.2.
Worked exampleExample
Question: Find the point where x + y = 6 and x − y = 2 meet on the graph.
Formula: The crossing point is the unique solution. Adding gives 2x = (x + y) + (x − y).
Substitution: 2x = 6 + 2 = 8, so x = 4. Then 4 + y = 6, so y = 2.
Answer: The lines meet at (4, 2). Check: 4 + 2 = 6 and 4 − 2 = 2. Exact solution x = 4, y = 2.
10-second revision
- Crossing = one solution = consistent
- Parallel = no solution = inconsistent
- Coincident = infinitely many = dependent and consistent
Board tip · BSEBBoard tip
On a BSEB graph write the scale of both axes and name the point of intersection.
Board tip · CBSEBoard tip
CBSE asks for a check. Put the meeting point back into both original equations.
Check your understandingall correct = mastery ★
1
A pair whose lines are parallel has —
💡 Hint:
The lines never meet.
Explanation:
No solution. The pair is inconsistent.
2
A pair of coincident lines has infinitely many solutions.
Explanation:
True — every point of the line fits both equations.
3
A pair with no solution is called an ______ pair.
💡 Hint:
The opposite of consistent.
Explanation:
Inconsistent.
4
On a graph a unique solution shows when the lines —
💡 Hint:
The panel with the red point.
Explanation:
Crossing at one point gives exactly one solution.
5
A dependent pair is always inconsistent.
Explanation:
False — a dependent pair is consistent, because there are infinitely many solutions.
6
Find the intersection of x + y = 4 and x − y = 0. Write the formula, the substitution and the check.
Model answer (board format)
The adding formula gives 2x = 4 + 0 = 4, so x = 2. Then y = 2. The point is (2, 2). Check: 2 + 2 = 4 and 2 − 2 = 0.
Check your answer:
The substitution method
प्रतिस्थापन विधि · NCERT 3.3.1 · Exercise 3.2 (current book)
Solve for one variable and put it inNotes
Choose the equation from which one variable comes out easily. Put that value into the other equation. A linear equation in one variable remains. Solve it, then put the first variable back.
A check in both original equations is required at the end. In the current NCERT this is exercise 3.2. In the older Bihar book it is exercise 3.3. Do not flip a sign in the middle step.
Write y in terms of x and substituteActivity
From x + y = 9 write y = 9 − x. If the second equation is 2x + y = 12, put 9 − x in place of y. After simplifying you get x, then y. Put the answer back into both lines and see that the two sides match.
Worked exampleExample
Question: Solve x + y = 9 and 2x + y = 12 by substitution.
Formula: From the first, y = 9 − x, then put it into the second.
Substitution: 2x + (9 − x) = 12, so x + 9 = 12 and x = 3. Then y = 9 − 3 = 6.
Answer: Exact solution x = 3, y = 6. Check: 3 + 6 = 9 and 6 + 6 = 12.
10-second revision
- Take one variable from one equation
- Put it into the other equation
- Check both original equations
Board tip · BSEBBoard tip
In a BSEB answer the step y = ... should stand on its own line. Do not jump to the final pair.
Board tip · CBSEBoard tip
CBSE asks for a numerical check after substitution. Write both equations.
Check your understandingall correct = mastery ★
1
Putting y = 2x into x + y = 12 gives x equal to —
💡 Hint:
x + 2x = 12.
Explanation:
3x = 12, x = 4 and y = 8.
2
After substitution, a check in only one equation is enough.
Explanation:
False — check both original equations.
3
From x + y = 9, y = ______ − x.
💡 Hint:
y alone on the left.
Explanation:
y = 9 − x.
4
Solve x − y = 1 and x + y = 7 by substitution.
Model answer (board format)
From the first, x = y + 1. In the second: y + 1 + y = 7, 2y = 6, y = 3, x = 4. Check: 4 − 3 = 1 and 4 + 3 = 7.
Check your answer:
The elimination method
विलोपन विधि · NCERT 3.3.2 · Exercise 3.3 (current book)
Match a coefficient and remove one variableNotes
Multiply the equations by numbers so that one variable has equal or opposite coefficients. Then add or subtract. That variable drops out.
The multiplication is of the whole line, not of one term only. The last algebra exercise of the current NCERT, 3.3, is this method. In the older book it is exercise 3.4. Put the remaining variable back into the first line.
Make the coefficients matchActivity
Take 2x + 3y = 13 and 3x − 2y = 0. Multiply the first by 2 and the second by 3, so the coefficients of y become +6 and −6. Adding removes y. Choose subtraction when both coefficients have the same sign.
Worked exampleExample
Question: Solve 2x + 3y = 13 and 3x − 2y = 0 by elimination.
Formula: First × 2 and second × 3, then add, so the y terms cancel.
Substitution: 4x + 6y = 26 and 9x − 6y = 0. Add: 13x = 26, x = 2. Then 3(2) − 2y = 0, y = 3.
Answer: Exact solution x = 2, y = 3. Check: 4 + 9 = 13 and 6 − 6 = 0.
10-second revision
- Multiply the whole line
- Subtract if the signs match, add if they are opposite
- Put the remaining variable into one original equation
Board tip · BSEBBoard tip
In a BSEB answer write both new lines after the multiplication. Do not jump straight to 13x = 26.
Board tip · CBSEBoard tip
On CBSE, if the coefficients already match, do not multiply without need.
Check your understandingall correct = mastery ★
1
Adding x + y = 8 and x − y = 2 gives x equal to —
💡 Hint:
y cancels and 2x remains.
Explanation:
2x = 10, x = 5 and y = 3.
2
In elimination it is correct to multiply only one term and leave the rest.
Explanation:
False — the whole equation is multiplied.
3
For x + y = 8 and x − y = 2, the solution has y = ______.
💡 Hint:
After x = 5, put it in the first.
Explanation:
y = 3.
4
If the coefficients of y are +4 and −4, to remove y you —
💡 Hint:
Opposite signs cancel on adding.
Explanation:
Adding cancels +4y and −4y.
5
Solve 3x + y = 11 and 3x − y = 7 by elimination.
Model answer (board format)
Add: 6x = 18, x = 3. Then 9 + y = 11, y = 2. Check: 9 + 2 = 11 and 9 − 2 = 7.
Check your answer:
Consistent or inconsistent from the ratios
अनुपात से संगत और असंगत · a1/a2 , b1/b2 , c1/c2
Three ratios, three casesNotes
For a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, comparing coefficients decides the case.
| Condition | Lines | Solution |
|---|---|---|
| a1/a2 ≠ b1/b2 | Intersect | One |
| a1/a2 = b1/b2 = c1/c2 | Coincident | Infinitely many |
| a1/a2 = b1/b2 ≠ c1/c2 | Parallel | None |
a1b2 − a2b1 ≠ 0 is the same statement as a1/a2 ≠ b1/b2. If some coefficient is zero, write this difference instead of a ratio. Take the sign of c from the standard form.
Standard form first, then the three ratiosActivity
Write 2x + 3y = 6 as 2x + 3y − 6 = 0. Write 4x + 6y = 12 as 4x + 6y − 12 = 0. Now 2/4, 3/6 and −6/−12 are all 1/2. The case is infinitely many solutions. Do not spoil the ratio by keeping c as +6.
Worked exampleExample
Question: For x + y − 4 = 0 and x − y − 2 = 0, name the case from the ratios, then solve.
Formula: a1/a2 = 1/1 = 1 and b1/b2 = 1/(−1) = −1. These are not equal, so there is one solution.
Substitution: Adding gives 2x − 6 = 0, x = 3. Then 3 + y − 4 = 0, y = 1.
Answer: Unique solution x = 3, y = 1. Check: 3 − 1 − 2 = 0.
10-second revision
- If the first two ratios are unequal, one solution
- If all three are equal, infinitely many
- If only the c ratio differs, no solution
Board tip · BSEBBoard tip
In a BSEB answer write all three ratios as fractions, then the name of the case.
Board tip · CBSEBoard tip
On CBSE, if a2 = 0 do not divide. Test the case with a1b2 − a2b1.
Check your understandingall correct = mastery ★
1
a1/a2 = b1/b2 ≠ c1/c2 means —
💡 Hint:
The lines are parallel.
Explanation:
No solution. The pair is inconsistent.
2
2x + 3y − 6 = 0 and 4x + 6y − 12 = 0 have infinitely many solutions.
Explanation:
True — all three ratios are 1/2.
3
The condition for one solution is a1/a2 ______ b1/b2.
💡 Hint:
Not equal.
Explanation:
a1/a2 ≠ b1/b2.
4
x + y = 2 and x + y = 5 are —
💡 Hint:
The ratios of a and of b match, not of c.
Explanation:
c1/c2 = −2/−5 = 2/5, which is not 1. No solution.
5
a1b2 − a2b1 = 0 always means infinitely many solutions.
Explanation:
False — it can be either parallel or coincident. Look at the ratio of c.
6
Decide the case of 3x + 2y − 5 = 0 and 6x + 4y − 10 = 0 from the ratios.
Model answer (board format)
3/6 = 2/4 = −5/−10 = 1/2. All three are equal, so the lines coincide and there are infinitely many solutions.
Check your answer:
Cross-multiplication — the older Bihar book
वज्र-गुणन — पुरानी बिहार किताब · Bihar 3.4.3 · Exercise 3.5 · not in the current NCERT
The formula, only when the denominator is not zeroNotes
The current NCERT exercise 3.3 stops at elimination. The older Bihar book keeps cross-multiplication in section 3.4.3. For a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0,
x / (b1c2 − b2c1) = y / (c1a2 − c2a1) = 1 / (a1b2 − a2b1)
There is one solution only when the denominator a1b2 − a2b1 is not zero. If it is zero, the ratio test decides between infinitely many solutions and none.
Test the denominator firstActivity
Copy the coefficients: under the first line put b1, c1, a1 and under the second b2, c2, a2. Each numerator is a cross difference. If the denominator is zero, do not divide by the formula. Call the same pair inconsistent or dependent from the ratios.
Worked exampleExample
Question: Solve x + y − 5 = 0 and x − y − 1 = 0 by cross-multiplication.
Formula: x = (b1c2 − b2c1) / (a1b2 − a2b1), y = (c1a2 − c2a1) / (a1b2 − a2b1).
Substitution: a1 = 1, b1 = 1, c1 = −5, a2 = 1, b2 = −1, c2 = −1. Denominator = 1·(−1) − 1·1 = −2. x = [1·(−1) − (−1)·(−5)] / (−2) = (−6)/(−2) = 3. y = [(−5)·1 − (−1)·1] / (−2) = 2.
Answer: Exact solution x = 3, y = 2. Check: 3 + 2 − 5 = 0 and 3 − 2 − 1 = 0.
10-second revision
- The denominator is a1b2 − a2b1
- The numerator of x is b1c2 − b2c1
- The numerator of y is c1a2 − c2a1
Board tip · BSEBBoard tip
The older BSEB book asks this method in exercise 3.5. If the denominator is zero, do not write a division.
Board tip · CBSEBoard tip
The current CBSE exercises 3.1 to 3.3 do not include cross-multiplication. Write the graph, substitution or elimination there.
Check your understandingall correct = mastery ★
1
In cross-multiplication the denominator of x is —
💡 Hint:
y has the same denominator.
Explanation:
The denominator is a1b2 − a2b1.
2
If the denominator is zero, cross-multiplication can still divide.
Explanation:
False — there is no division by zero. Decide the case from the ratios.
3
The numerator of x is b1c2 − ______.
💡 Hint:
The product of the second b and the first c.
Explanation:
b2c1.
4
Solve x + y − 3 = 0 and x − y − 1 = 0 by cross-multiplication.
Model answer (board format)
a1=1, b1=1, c1=−3, a2=1, b2=−1, c2=−1. Denominator = −1 − 1 = −2. x = [1·(−1) − (−1)·(−3)]/(−2) = (−4)/(−2) = 2. y = [(−3)·1 − (−1)·1]/(−2) = 1. Check: 2 + 1 − 3 = 0.
Check your answer:
❓ Full question bank — with answers and explanations — 64 questions
Pick a type. The 30 lesson checks are separate — each lesson has as many as its topic needs. All correct earns mastery ★.
BSEB modelpractice · not annual60 tried · 0 correctMultiple choiceMCQ120 tried · 0 correctTrue / falseTrue / false80 tried · 0 correctFill in the blanksFill80 tried · 0 correctMatchMatch20 tried · 0 correctAssertion–reasonAssertion–reason50 tried · 0 correctBalancingBalancing40 tried · 0 correctIdentifyIdentify20 tried · 0 correctShort and longVSA, SA, LA110 tried · 0 correctCBSE-stylenot a PYQ60 tried · 0 correct
No question is marked as a verified past paper. The BSEB set is a model for practice. CBSE items are CBSE-style, not a copy of any year’s paper.
Multiple choice
0/12Pick one option. A wrong try brings a hint.
1
In standard form, c for x + 2y = 11 is —
💡 Hint:
The constant moves left.
Explanation:
x + 2y − 11 = 0, so c = −11.
2
The correct condition for one solution is —
💡 Hint:
The first two ratios differ.
Explanation:
When a1/a2 ≠ b1/b2 the lines cross at one point.
3
The lines of x + y = 1 and x + y = 9 —
💡 Hint:
The ratios of a and of b match.
Explanation:
The ratio of c differs, so they are parallel and there is no solution.
4
Putting y = 1 into 2x + y = 7 gives x as —
💡 Hint:
2x + 1 = 7.
Explanation:
2x = 6, x = 3.
5
A dependent pair is —
💡 Hint:
Infinitely many solutions are still consistent.
Explanation:
A dependent pair is consistent.
6
In elimination, equal coefficients with opposite signs are removed by —
💡 Hint:
+4y and −4y.
Explanation:
They cancel on adding.
7
The solution of y = x + 1 and x + y = 5 is —
💡 Hint:
x + (x + 1) = 5.
Explanation:
2x = 4, x = 2, y = 3.
8
a1b2 − a2b1 ≠ 0 means —
💡 Hint:
This is a1/a2 ≠ b1/b2.
Explanation:
One unique solution.
9
On coincident lines the solutions are —
💡 Hint:
Every point of the line.
Explanation:
Infinitely many solutions.
10
4x − 2y = 0 and 2x − y = 0 are —
💡 Hint:
The first is twice the second.
Explanation:
All three ratios match, so there are infinitely many solutions.
11
On the graph the scale of the two axes should be —
💡 Hint:
Otherwise the point is misread.
Explanation:
Keep the same scale.
12
In cross-multiplication the numerator of y is —
💡 Hint:
The numerator of x is the other one.
Explanation:
The numerator of y is c1a2 − c2a1.
True or false
0/81
Every linear pair has exactly one solution.
Explanation:
False — there may be no solution or infinitely many.
2
Parallel lines show an inconsistent pair.
Explanation:
True — there is no intersection.
3
In x + y = 5 the coefficient of y is 5.
Explanation:
False — the coefficient of y is 1 and c = −5.
4
Cancelling one equation always leaves an equivalent pair.
Explanation:
False — both equations have to stay.
5
If a1/a2 = b1/b2 = c1/c2, the lines coincide.
Explanation:
True — infinitely many solutions.
6
Substitution and elimination can give different solutions when the arithmetic is correct.
Explanation:
False — correct arithmetic gives the same solution in both.
7
An inconsistent pair is not consistent.
Explanation:
True — there is no solution.
8
Cross-multiplication is exercise 3.3 of the current NCERT.
Explanation:
False — 3.3 is elimination. Cross-multiplication is exercise 3.5 of the older Bihar book.
Fill in the blanks
0/81
Lines that never meet are called ______.
💡 Hint:
Parallel.
Explanation:
Same slope, different place.
2
A pair with infinitely many solutions is called ______.
💡 Hint:
Dependent.
Explanation:
It is also consistent.
3
In a1x + b1y + c1 = 0, c1 is the ______ term.
💡 Hint:
Constant.
Explanation:
It is not with x or y.
4
For a unique solution the lines ______ each other.
💡 Hint:
They intersect.
Explanation:
One point.
5
If there is no solution, the number of solutions is ______.
💡 Hint:
Zero.
Explanation:
The parallel case.
6
In elimination the whole ______ is multiplied.
💡 Hint:
The equation.
Explanation:
Not one term.
7
In x + y = 6 and x − y = 2, x = ______.
💡 Hint:
4.
Explanation:
Adding gives 2x = 8.
8
If a1/a2 = b1/b2 ≠ c1/c2, the pair is ______.
💡 Hint:
Inconsistent.
Explanation:
Parallel lines.
Match
0/21
Match the case with its picture.
Column B: A. Parallel lines · B. Intersecting lines · C. Coincident lines · D. No intersection
1. One solution
2. No solution
3. Infinitely many
4. Inconsistent
Explanation:
One solution is a crossing, none is parallel, infinitely many is coincident. Inconsistent also has no intersection.
2
Match the method with its first step.
Column B: A. Join two points · B. Write one variable in terms of the other · C. Write the denominator a1b2 − a2b1 · D. Match a coefficient and add
1. Substitution
2. Elimination
3. Graph
4. Cross-multiplication
Explanation:
Substitution rewrites a variable, elimination matches a coefficient, the graph joins points, and cross-multiplication writes the denominator.
Assertion–reason
0/5Check both statements, then see whether the reason explains the assertion.
1
Assertion (A): x + y = 2 and x + y = 5 have no solution.
Reason (R): a1/a2 = b1/b2 but c1/c2 differs, so the lines are parallel.
Explanation:
Both are true and R is the reason for A.
2
Assertion (A): If a1/a2 ≠ b1/b2 there is exactly one solution.
Reason (R): The graph of one linear equation is a straight line.
Explanation:
Both are true, but R does not explain the ratio condition.
3
Assertion (A): 2x + 4y = 8 and x + 2y = 4 have infinitely many solutions.
Reason (R): These lines cross at exactly one point.
Explanation:
A is true because the first is twice the second. R is false.
4
Assertion (A): Substitution means the pair is inconsistent.
Reason (R): An inconsistent pair has no common solution.
Explanation:
A is false. Substitution is a method. R is true.
5
Assertion (A): Coincident lines give a consistent pair.
Reason (R): Every point of the line is a solution of both equations.
Explanation:
Both are true and R explains A.
Coefficient practice
0/4These chemistry lines are not a result of this maths chapter. They are only practice in filling coefficients. A blank means 1.
1
This is coefficient practice and not a result of this maths chapter. In the line where H2 + O2 makes water, Fill coefficients (blank = 1).
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
2
This is coefficient practice and not a result of this maths chapter. In the line where N2 + H2 makes ammonia, Fill coefficients (blank = 1).
N2 + H2 → NH3
Explanation: N2 + 3H2 → 2NH3
3
This is coefficient practice and not a result of this maths chapter. In the line where C + O2 makes carbon dioxide, Fill coefficients (blank = 1).
C + O2 → CO2
Explanation: C + O2 → CO2
4
This is coefficient practice and not a result of this maths chapter. In the burning line of CH4 + O2, Fill coefficients (blank = 1).
CH4 + O2 → CO2 + H2O
Explanation: CH4 + 2O2 → CO2 + 2H2O
Classify
0/21
Place each pair by how many solutions it has.
x + y = 4 and x − y = 2
x + y = 2 and x + y = 5
2x + 3y = 6 and 4x + 6y = 12
x + 2y = 3 and 2x + 4y = 10
Explanation:
The first crosses at (3, 1). The second and the fourth are parallel. The third is double the first equation.
2
Place each sentence as consistent or inconsistent.
The lines cross at one point
The lines are parallel
The lines coincide
a1/a2 = b1/b2 ≠ c1/c2
Explanation:
Crossing and coinciding are consistent. The parallel case is inconsistent.
Very short answer
0/41
Define a consistent pair in one line.
Model answer (board format)
A pair with at least one solution is consistent.
Check your answer:
2
Write x − y = 3 in standard form.
Model answer (board format)
x − y − 3 = 0. a = 1, b = −1, c = −3.
Check your answer:
3
What is a pair of parallel lines called, and how many solutions does it have?
Model answer (board format)
It is called inconsistent. There are zero solutions.
Check your answer:
4
Write the denominator of cross-multiplication.
Model answer (board format)
a1b2 − a2b1. There is one solution only when this is not zero.
Check your answer:
Short answer
0/41
Solve 2x + y = 10 and x + y = 6 by substitution.
Model answer (board format)
From the second, y = 6 − x. In the first: 2x + 6 − x = 10, x = 4, y = 2. Check: 8 + 2 = 10 and 4 + 2 = 6.
Check your answer:
2
Solve 5x + 2y = 16 and 5x − 2y = 4 by elimination.
Model answer (board format)
Add: 10x = 20, x = 2. Then 10 + 2y = 16, y = 3. Check: 10 − 6 = 4.
Check your answer:
3
State the case of x + 2y = 7 and 2x + 4y = 14. Why does a search for one solution stop?
Model answer (board format)
1/2 = 2/4 = 7/14. All three ratios are equal, so there are infinitely many solutions. One single pair cannot be chosen.
Check your answer:
4
Write the three graphical cases and their names.
Model answer (board format)
Crossing: one solution, consistent. Parallel: no solution, inconsistent. Coincident: infinitely many, dependent and consistent.
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Long answer
0/31
Solve x + y = 10 and x − y = 4 by both substitution and elimination. Match the two answers.
Model answer (board format)
Substitution: x = 10 − y, then 10 − y − y = 4, 10 − 2y = 4, y = 3, x = 7. Elimination: adding gives 2x = 14, x = 7, then y = 3. Both give the exact solution x = 7, y = 3. Check: 7 + 3 = 10 and 7 − 3 = 4.
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2
Decide the case of these three pairs from the ratios: (i) x + y = 5, x − y = 1 (ii) x + y = 5, 2x + 2y = 10 (iii) x + y = 5, x + y = 8.
Model answer (board format)
(i) 1/1 ≠ 1/(−1), one solution. Adding gives x = 3, y = 2. (ii) 1/2 = 1/2 = 5/10, infinitely many. (iii) 1/1 = 1/1 ≠ 5/8, no solution.
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3
Solve 2x + y − 7 = 0 and x − y − 2 = 0 by cross-multiplication. Which book’s exercise contains this method?
Model answer (board format)
a1=2, b1=1, c1=−7, a2=1, b2=−1, c2=−2. Denominator = 2·(−1) − 1·1 = −3. x = [1·(−2) − (−1)·(−7)]/(−3) = (−9)/(−3) = 3. y = [(−7)·1 − (−2)·2]/(−3) = (−3)/(−3) = 1. Check: 6 + 1 − 7 = 0 and 3 − 1 − 2 = 0. This is exercise 3.5 of the older Bihar book. The current NCERT stops at elimination in 3.3.
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BSEB model paper · practice
0/6This model set is for practice. It is not a question from any year’s annual examination.
1
Which of these is an inconsistent pair?
💡 Hint:
Same left side, different right side.
Explanation:
x + y = 3 and x + y = 6 are parallel.
2
In x + y = 9 and x − y = 1, y is —
💡 Hint:
Subtraction leaves 2y.
Explanation:
2y = 8, y = 4 and x = 5.
3
If a1/a2 = b1/b2 = c1/c2, the solutions are ______.
💡 Hint:
Infinitely many.
Explanation:
Coincident lines.
4
A student wrote x + y = 8 as x + y + 8 = 0. What is the mistake? Write the correct c.
Model answer (board format)
The sign of the constant was not changed. The correct form is x + y − 8 = 0, so c = −8.
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5
Solve 3x + 2y = 12 and x − y = 1 by elimination. Write the steps and the check.
Model answer (board format)
Second × 2: 2x − 2y = 2. Add to the first: 5x = 14, x = 14/5. Then y = 14/5 − 1 = 9/5. Check: 3·(14/5) + 2·(9/5) = 60/5 = 12.
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6
Assertion (A): Intersecting lines have one single solution.
Reason (R): Parallel lines also meet at one point.
Explanation:
A is true. R is false, parallel lines do not meet.
CBSE-style questions
0/6These are competency-based practice questions. They are not a copy of any year’s paper.
1
A student used different scales and read the intersection as (2, 3), while the equations are x + y = 7 and x − y = 1. The correct solution is —
💡 Hint:
Adding gives 2x = 8.
Explanation:
x = 4, y = 3. (2, 3) fails the check because 2 + 3 = 5 ≠ 7.
2
Two pens and one eraser cost 14 rupees. One pen and one eraser cost 9 rupees. One pen costs —
💡 Hint:
Subtraction leaves one pen.
Explanation:
(2p + e) − (p + e) = 14 − 9, so p = 5 rupees.
3
Assertion (A): x/2 + y/3 = 1 and 3x + 2y = 6 are the same line.
Reason (R): Multiplying the first by 6 gives 3x + 2y = 6.
Explanation:
Both are true and R explains A. Infinitely many solutions.
4
Assertion (A): a1/a2 = b1/b2 is enough for infinitely many solutions.
Reason (R): For infinitely many solutions, c1/c2 must equal them too.
Explanation:
A is false, because if c differs there is no solution. R is true.
5
A notebook says that x + 2y = 4 and 2x + 4y = 9 has the solution x = 1, y = 1.5. Check it and state the case.
Model answer (board format)
1 + 2·1.5 = 4 fits the first, but 2 + 4·1.5 = 8 ≠ 9. The ratios are 1/2 = 2/4 ≠ 4/9. The pair is inconsistent and has no solution.
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6
Which methods are exercises 3.2 and 3.3 of the current book? What is 3.5 of the older book? Write the difference in one sentence.
Model answer (board format)
In the current NCERT, 3.2 is substitution and 3.3 is elimination. Exercise 3.5 of the older Bihar book is cross-multiplication. Current CBSE practice does not ask for cross-multiplication.
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🏛️ Board exam corner — Bihar Board (BSEB)— CBSE
Switch board with BSEB | CBSE above. The lessons follow the same NCERT chapter.
BSEB model · practiceBoard tip
This page has no verified annual-exam question, because no source page has been added. The model set below is practice in the board pattern.
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CBSE-style · competency-based
These are case and assertion-reason practice items. Do not treat them as past CBSE questions.
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🧠 What you learned + equation sheet
What you learned
- A pair is two linear equations; a solution is an (x, y) that fits both.
- Crossing lines have one solution, parallel lines none, coincident lines infinitely many.
- a1/a2 ≠ b1/b2 means one solution; if all three ratios match, there are infinitely many.
- a1/a2 = b1/b2 ≠ c1/c2 is an inconsistent pair. Write the sign of the constant with care.
- In substitution, put one variable in; in elimination, add or subtract to remove one.
- Cross-multiplication is in the older Bihar book. The current NCERT exercise 3.3 ends at elimination.
| What | Keep this |
|---|---|
| One solution | a1/a2 ≠ b1/b2 |
| No solution | a1/a2 = b1/b2 ≠ c1/c2 |
| Infinitely many | a1/a2 = b1/b2 = c1/c2 |
| Substitution | place one variable into the other equation |
| Elimination | add or subtract after matching a coefficient |
| Cross-multiplication | older Bihar 3.4.3; not in current NCERT exercises |
The notes are original writing. The textbook was used only for activity order and numbers. “Verified” will be used only on a question that has a source page.