Exercise 3.1 answers: Pair of Linear Equations in Two Variables

Open NCERT p. 28Matches NCERT’s answer

(i) 10 students of Class X took part in a Mathematics quiz.

1. Let the number of boys be x and the number of girls be y.
2. Total students: x + y = 10.
3. Girls are 4 more than boys: y = x + 4.
4. Points for x + y = 10: x = 0 gives y = 10, and x = 10 gives y = 0. Plot (0, 10) and (10, 0) and draw the line through them.
5. Points for y = x + 4: x = 0 gives y = 4, and x = 2 gives y = 6. Plot (0, 4) and (2, 6) and draw the line through them.
6. The two lines cross at (3, 7), so x = 3 and y = 7.
7. Check in both equations: 3 + 7 = 10, and 3 + 4 = 7.

Answer3 boys and 7 girls took part in the quiz.

(ii) 5 pencils and 7 pens together cost ₹ 50

1. Let one pencil cost ₹x and one pen cost ₹y.
2. 5 pencils and 7 pens cost ₹50: 5x + 7y = 50.
3. 7 pencils and 5 pens cost ₹46: 7x + 5y = 46.
4. Points for 5x + 7y = 50: x = 10 gives y = 0, and x = −4 gives y = 10. Plot (10, 0) and (−4, 10) and draw the line.
5. Points for 7x + 5y = 46: x = 8 gives y = −2, and x = −2 gives y = 12. Plot (8, −2) and (−2, 12) and draw the line.
6. The two lines cross at (3, 5), so x = 3 and y = 5.
7. Check in both equations: 5(3) + 7(5) = 15 + 35 = 50, and 7(3) + 5(5) = 21 + 25 = 46.

AnswerOne pencil costs ₹3 and one pen costs ₹5.

Watch this explained “One sentence, one condition”, 1:27 into Why a pair of these equations is a pair of straight lines

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(i) 5x − 4y + 8 = 0; 7x + 6y − 9 = 0

1. a1/a2 = 5/7. b1/b2 = −4/6 = −2/3.
2. 5/7 is not equal to −2/3, so the lines are not parallel or coincident.

AnswerThe lines intersect at a point.

(ii) 9x + 3y + 12 = 0; 18x + 6y + 24 = 0

1. a1/a2 = 9/18 = 1/2. b1/b2 = 3/6 = 1/2. c1/c2 = 12/24 = 1/2.
2. All three ratios are equal.

AnswerThe lines are coincident.

(iii) 6x − 3y + 10 = 0; 2x − y + 9 = 0

1. a1/a2 = 6/2 = 3. b1/b2 = −3/−1 = 3.
2. c1/c2 = 10/9, which is not 3.
3. The first two ratios are equal but the third is different.

AnswerThe lines are parallel.

Watch this explained “The three comparisons”, 1:29 into Predicting which of the three you will get by comparing coefficients

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(i) 3x + 2y = 5; 2x − 3y = 7

1. a1/a2 = 3/2. b1/b2 = 2/−3.
2. These are not equal, so the lines cross once.

AnswerConsistent.

(ii) 2x − 3y = 8; 4x − 6y = 9

1. a1/a2 = 2/4 = 1/2. b1/b2 = −3/−6 = 1/2. c1/c2 = 8/9.
2. The first two ratios match but the third does not — the lines are parallel.

AnswerInconsistent.

(iii) 9x − 10y = 14

1. a1/a2 = (3/2)/9 = 1/6. b1/b2 = (5/3)/(−10) = −1/6.
2. These are not equal, so the lines cross once.

AnswerConsistent.

(iv) 5x − 3y = 11; −10x + 6y = −22

1. a1/a2 = 5/−10 = −1/2. b1/b2 = −3/6 = −1/2. c1/c2 = 11/−22 = −1/2.
2. All three ratios are equal — one equation is the other one scaled.

AnswerConsistent (in fact, the equations give the same line, so there are infinitely many solutions).

(v) 2x + 3y = 12

1. a1/a2 = (4/3)/2 = 2/3. b1/b2 = 2/3. c1/c2 = 8/12 = 2/3.
2. All three ratios are equal — one equation is the other one scaled.

AnswerConsistent (the equations give the same line, so there are infinitely many solutions).

Watch this explained “What we have”, 14:23 into Predicting which of the three you will get by comparing coefficients

Open NCERT p. 29Checked by computer

(i) x + y = 5, 2x + 2y = 10

1. Write the pair as x + y − 5 = 0 and 2x + 2y − 10 = 0.
2. a1/a2 = 1/2, b1/b2 = 1/2, c1/c2 = −5/−10 = 1/2. All three ratios are equal, so the lines coincide and the pair is consistent.
3. Graph: x + y = 5 passes through (0, 5) and (5, 0). The second equation, 2x + 2y = 10, is the first one doubled and passes through the same two points, so both equations draw the same line.
4. Every point on this line is a solution, for example (0, 5), (5, 0) or (2, 3).

AnswerConsistent, with infinitely many solutions: every point on the line x + y = 5.

(ii) x − y = 8, 3x − 3y = 16

1. Write the pair as x − y − 8 = 0 and 3x − 3y − 16 = 0.
2. a1/a2 = 1/3, b1/b2 = −1/−3 = 1/3, c1/c2 = −8/−16 = 1/2.
3. The first two ratios are equal but the third is different, so the lines are parallel and never meet.

AnswerInconsistent: no solution.

(iii) 2x + y − 6 = 0, 4x − 2y − 4 = 0

1. a1/a2 = 2/4 = 1/2 and b1/b2 = 1/−2 = −1/2. These are different, so the lines cross at exactly one point and the pair is consistent.
2. Points for 2x + y − 6 = 0: x = 0 gives y = 6, and x = 3 gives y = 0. Plot (0, 6) and (3, 0) and draw the line.
3. Points for 4x − 2y − 4 = 0: x = 0 gives y = −2, and x = 1 gives y = 0. Plot (0, −2) and (1, 0) and draw the line.
4. The two lines cross at (2, 2).
5. Check: 2(2) + 2 − 6 = 0 and 4(2) − 2(2) − 4 = 0.

AnswerConsistent, with the solution x = 2, y = 2.

(iv) 2x − 2y − 2 = 0, 4x − 4y − 5 = 0

1. a1/a2 = 2/4 = 1/2, b1/b2 = −2/−4 = 1/2, c1/c2 = −2/−5 = 2/5.
2. The first two ratios are equal but the third is different, so the lines are parallel and never meet.

AnswerInconsistent: no solution.

Watch this explained “Drawing a pair properly”, 7:20 into Crossing, parallel or lying on top: what each picture says about solutions

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1. Let the width be x m and the length be y m.
2. Length is 4 m more than width: y = x + 4.
3. Half the perimeter is length + width, so x + y = 36.
4. Put y = x + 4 into x + y = 36: x + (x + 4) = 36, so 2x = 32, x = 16.
5. Then y = 16 + 4 = 20.

AnswerThe garden is 20 m long and 16 m wide.

Watch this explained “One sentence, one condition”, 1:27 into Why a pair of these equations is a pair of straight lines

Open NCERT p. 29Checked by computerAnswers can differ: one example

(i) intersecting lines

1. For intersecting lines we need a1/a2 ≠ b1/b2. Any equation whose x and y coefficients are not in the ratio 2 : 3 will do, so there are many correct answers. Here is one.
2. Take x + y − 5 = 0. Then a1/a2 = 2/1 = 2 and b1/b2 = 3/1 = 3.
3. 2 ≠ 3, so the lines intersect.

Answerx + y − 5 = 0 (one of many possible answers)

(ii) parallel lines

1. For parallel lines we need a1/a2 = b1/b2 ≠ c1/c2. Multiply the x and y terms of 2x + 3y by the same number, but choose any constant that does not fit that same multiple. Many answers are possible. Here is one.
2. Doubling the x and y terms gives 4x + 6y; doubling −8 would give −16, so choose a different constant, such as +7: 4x + 6y + 7 = 0.
3. a1/a2 = 2/4 = 1/2, b1/b2 = 3/6 = 1/2, c1/c2 = −8/7. The first two are equal and the third is not, so the lines are parallel.

Answer4x + 6y + 7 = 0 (one of many possible answers)

(iii) coincident lines

1. For coincident lines we need a1/a2 = b1/b2 = c1/c2. Multiply the whole equation, constant included, by any non-zero number. Many answers are possible. Here is one.
2. Doubling 2x + 3y − 8 = 0 gives 4x + 6y − 16 = 0.
3. a1/a2 = 2/4 = 1/2, b1/b2 = 3/6 = 1/2, c1/c2 = −8/−16 = 1/2. All three are equal, so the lines coincide.

Answer4x + 6y − 16 = 0 (one of many possible answers)

Watch this explained “Building a partner equation to order”, 10:09 into Predicting which of the three you will get by comparing coefficients

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1. Line 1: x − y + 1 = 0, that is y = x + 1. Points: x = −1 gives y = 0, x = 0 gives y = 1, x = 2 gives y = 3. Plot (−1, 0), (0, 1), (2, 3) and draw the line.
2. Line 2: 3x + 2y − 12 = 0, that is y = (12 − 3x)/2. Points: x = 4 gives y = 0, x = 0 gives y = 6, x = 2 gives y = 3. Plot (4, 0), (0, 6), (2, 3) and draw the line.
3. Both lines pass through (2, 3), so that is where they meet. Check: 2 − 3 + 1 = 0 and 3(2) + 2(3) − 12 = 0.
4. Line 1 meets the x-axis (where y = 0) at (−1, 0), and Line 2 meets it at (4, 0).
5. The two lines and the x-axis enclose a triangle with these three corners. Shade the region inside it.

AnswerThe vertices of the triangle are (−1, 0), (4, 0) and (2, 3).

Watch this explained “Drawing a pair properly”, 7:20 into Crossing, parallel or lying on top: what each picture says about solutions