Pair of linear equations

Q: Which method is best: substitution or elimination?

Substitution is quickest when one equation already gives x or y on its own. Elimination is quickest when the coefficients are easy to match.

A pair of linear equations in two variables, such as $x + y = 10$ and $x - y = 4$ , describes two straight lines. Solving them means finding the values of $x$ and $y$ that make both equations true: the point where the lines cross.

Formulas

General form

a_{1} x + b_{1} y + c_{1} = 0 a_{2} x + b_{2} y + c_{2} = 0

One solution (lines cross)

\frac{a _{1}}{a _{2}} \neq = \frac{b _{1}}{b _{2}}

No solution (parallel) / infinitely many (same line)

\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} \neq = \frac{c _{1}}{c _{2}} \frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}}

Worked examples

Example 1. Solve

x + y = 10

and

x - y = 4

by elimination.

Add the equations: $2 x = 14$ , so $x = 7$ .
Then $7 + y = 10$ , so $y = 3$ .

Answer:

x = 7, y = 3

Example 2. Solve

y = 2 x - 1

and

3 x + y = 9

by substitution.

Put $y = 2 x - 1$ into the second equation: $3 x + 2 x - 1 = 9$ .
$5 x = 10$ , so $x = 2$ and $y = 2 \times 2 - 1 = 3$ .

Answer:

x = 2, y = 3

Example 3. 2 pens and 3 notebooks cost ₹70; 1 pen and 2 notebooks cost ₹45. Find the cost of each.

Let a pen cost $p$ and a notebook $n : 2 p + 3 n = 70$ and $p + 2 n = 45$ .
From the second, $p = 45 - 2 n$ . Then $2 (45 - 2 n) + 3 n = 70$ , so $90 - n = 70$ and $n = 20$ .
$p = 45 - 40 = 5$ . Check: $2 \times 5 + 3 \times 20 = 70$ . ✓

Answer: Pen ₹5, notebook ₹20

Example 4. How many solutions do

2 x + 3 y = 5

and

4 x + 6 y = 7

have?

$\frac{2}{4} = \frac{3}{6} = \frac{1}{2}$ , but $\frac{5}{7} \neq = \frac{1}{2}$ .
The lines are parallel and never meet.

Answer: No solution

Common mistakes

Substituting back into the same equation you rearranged. Check the answer in BOTH original equations.
Adding when you should subtract in elimination. Make one variable's coefficients equal, then subtract if they have the same sign, add if opposite.
Forgetting to multiply every term, including the constant, when scaling an equation.

Practice questions with answers

1. Solve:

x + y = 10

and

x - y = 4

A $x = 7, y = 3$
B $x = 6, y = 4$
C $x = 5, y = 5$
D $x = 8, y = 2$

Show answer

Answer: A)

x = 7, y = 3

Add:

2 x = 14

→

x = 7

. Then

y = 10 - 7 = 3

2. In the elimination method, what is the first goal when solving a pair of linear equations?

AMake the coefficient of one variable equal (in magnitude) in both equations
BMake the constant terms equal
CSquare both equations
DFind the $y$ -intercept of each line

Show answer

Answer: A) Make the coefficient of one variable equal (in magnitude) in both equations

Elimination works by adding or subtracting the equations so that one variable cancels; this requires its coefficients to be equal in magnitude first.

3. If

y = 5

and

x + y = 9

, then

x

= ?

Show answer

Answer: D) 4

Substitute

y = 5

into

x + y = 9 : x + 5 = 9

, so

x = 4

4. The graph of a linear equation in two variables is always a:

AStraight line
BParabola
CCircle
DCurve

Show answer

Answer: A) Straight line

Any equation of the form ax + by + c = 0 (a, b not both zero) plots as a straight line.

5. The sum of two numbers is 20 and their difference is 4. What is the larger number?

Show answer

Answer: A) 12

Let numbers be

x

and

y . x + y = 20, x - y = 4

. Adding:

2 x = 24, x = 12

6. A woman does u and a man does v of a job per day. '2 women and 5 men finish in 4 days' gives which equation?

A $2 u + 5 v = 4$
B $2/ u + 5/ v = 4$
C $2 u + 5 v = 1/4$
D $u + v = 1/4$

Show answer

Answer: C)

2 u + 5 v = 1/4

One day's work of the team is

2 u + 5 v

; finishing in 4 days means it is

\frac{1}{4}

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Frequently asked questions

Which method is best: substitution or elimination?

Substitution is quickest when one equation already gives $x$ or $y$ on its own. Elimination is quickest when the coefficients are easy to match.

What does the graph of a pair of linear equations show?

Two straight lines. Crossing lines give one solution, parallel lines none, and the same line drawn twice gives infinitely many.

What is a consistent pair of equations?

A pair that has at least one solution (one, or infinitely many). An inconsistent pair has none.

How do I set up equations from a word problem?

Choose a letter for each unknown, then turn each sentence that gives a total or a relation into one equation.

Pair of linear equations

Formulas

Worked examples

Common mistakes

Practice questions with answers

Practice by class

Frequently asked questions

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