Sec 2 Maths · Linear equations, inequalities and simultaneous equations

Simultaneous equations: substitution questions with answers

Practice questions on simultaneous equations: substitution, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

Solve the simultaneous equations $-x + 3y = -20$ and $y = -4x + 28$ by substitution.

Step 1 of 4

Number the equations

-x + 3y = -20

... (1)

y = -4x + 28

... (2)

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Questions

Question 1One equation gives y: substitute it in a bracket
Solve the simultaneous equations $2x - 4y = 6$ and $y = 3x + 1$ by substitution.
Show the worked solution
1. Step 1: Number the equations $2x - 4y = 6$ ... (1) $y = 3x + 1$ ... (2)
2. Step 2: Substitute (2) into (1): put the whole expression for y in a BRACKET $2x - 4(3x + 1) = 6$
3. Step 3: Expand the bracket (the number in front multiplies BOTH terms), then collect $2x - 12x - 4 = 6$ $-10x = 10$ $x = \dfrac{10}{-10}$ $= -1$
4. Step 4: Put your value of x into (2) to find y $y = 3(-1) + 1$ $= -2$ Check in (1): $2(-1) - 4(-2)$ $= 6$ , the right-hand side of (1).
Question 2Make y the subject first
Solve the simultaneous equations $7x + 5y = 69$ and $5x - y = 31$ by substitution.
Show the worked solution
1. Step 1: Number the equations $7x + 5y = 69$ ... (1) $5x - y = 31$ ... (2)
2. Step 2: Make y the subject of (2): move the x-term across, then change every sign (divide by −1) $y = 5x - 31$ ... (3)
3. Step 3: Substitute (3) into (1): put the whole expression for y in a BRACKET $7x + 5(5x - 31) = 69$
4. Step 4: Expand the bracket (the number in front multiplies BOTH terms), then collect $7x + 25x - 155 = 69$ $32x = 224$ $x = \dfrac{224}{32}$ $= 7$
5. Step 5: Put your value of x into (3) to find y $y = 5(7) - 31$ $= 4$ Check in (1): $7(7) + 5(4)$ $= 69$ , the right-hand side of (1).
Question 3Make x the subject first
Solve the simultaneous equations $x - 6y = -50$ and $-x + 4y = 34$ by substitution.
Show the worked solution
1. Step 1: Number the equations $x - 6y = -50$ ... (1) $-x + 4y = 34$ ... (2)
2. Step 2: Make x the subject of (2): move the y-term across, then change every sign (divide by −1) $x = 4y - 34$ ... (3)
3. Step 3: Substitute (3) into (1): put the whole expression for x in a BRACKET $(4y - 34) - 6y = -50$
4. Step 4: Expand the bracket (the number in front multiplies BOTH terms), then collect $4y - 34 - 6y = -50$ $-2y = -16$ $y = \dfrac{-16}{-2}$ $= 8$
5. Step 5: Put your value of y into (3) to find x $x = 4(8) - 34$ $= -2$ Check in (1): $(-2) - 6(8)$ $= -50$ , the right-hand side of (1).
Question 4Where two lines meet
The lines $-5x + 6y = -14$ and $-x + 4y = -14$ meet at the point $P$ . Find the coordinates of $P$ .
Show the worked solution
1. Step 1: P is on both lines, so its x and y fit both equations: solve them together. Number the equations $-5x + 6y = -14$ ... (1) $-x + 4y = -14$ ... (2)
2. Step 2: Make x the subject of (2): move the y-term across, then change every sign (divide by −1) $x = 4y + 14$ ... (3)
3. Step 3: Substitute (3) into (1): put the whole expression for x in a BRACKET $-5(4y + 14) + 6y = -14$
4. Step 4: Expand the bracket (the number in front multiplies BOTH terms), then collect $-20y - 70 + 6y = -14$ $-14y = 56$ $y = \dfrac{56}{-14}$ $= -4$
5. Step 5: Put your value of y into (3) to find x $x = 4(-4) + 14$ $= -2$ Check in (1): $-5(-2) + 6(-4)$ $= -14$ , the right-hand side of (1).

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Questions

Question 1One equation gives y: substitute it in a bracket

Solve the simultaneous equations $2x - 4y = 6$ and $y = 3x + 1$ by substitution.

Show the worked solution

Step 1: Number the equations $2x - 4y = 6$ ... (1) $y = 3x + 1$ ... (2)
Step 2: Substitute (2) into (1): put the whole expression for y in a BRACKET $2x - 4(3x + 1) = 6$
Step 3: Expand the bracket (the number in front multiplies BOTH terms), then collect $2x - 12x - 4 = 6$ $-10x = 10$ $x = \dfrac{10}{-10}$ $= -1$
Step 4: Put your value of x into (2) to find y $y = 3(-1) + 1$ $= -2$ Check in (1): $2(-1) - 4(-2)$ $= 6$ , the right-hand side of (1).

Question 2Make y the subject first

Solve the simultaneous equations $7x + 5y = 69$ and $5x - y = 31$ by substitution.

Show the worked solution

Step 1: Number the equations $7x + 5y = 69$ ... (1) $5x - y = 31$ ... (2)
Step 2: Make y the subject of (2): move the x-term across, then change every sign (divide by −1) $y = 5x - 31$ ... (3)
Step 3: Substitute (3) into (1): put the whole expression for y in a BRACKET $7x + 5(5x - 31) = 69$
Step 4: Expand the bracket (the number in front multiplies BOTH terms), then collect $7x + 25x - 155 = 69$ $32x = 224$ $x = \dfrac{224}{32}$ $= 7$
Step 5: Put your value of x into (3) to find y $y = 5(7) - 31$ $= 4$ Check in (1): $7(7) + 5(4)$ $= 69$ , the right-hand side of (1).

Question 3Make x the subject first

Solve the simultaneous equations $x - 6y = -50$ and $-x + 4y = 34$ by substitution.

Show the worked solution

Step 1: Number the equations $x - 6y = -50$ ... (1) $-x + 4y = 34$ ... (2)
Step 2: Make x the subject of (2): move the y-term across, then change every sign (divide by −1) $x = 4y - 34$ ... (3)
Step 3: Substitute (3) into (1): put the whole expression for x in a BRACKET $(4y - 34) - 6y = -50$
Step 4: Expand the bracket (the number in front multiplies BOTH terms), then collect $4y - 34 - 6y = -50$ $-2y = -16$ $y = \dfrac{-16}{-2}$ $= 8$
Step 5: Put your value of y into (3) to find x $x = 4(8) - 34$ $= -2$ Check in (1): $(-2) - 6(8)$ $= -50$ , the right-hand side of (1).

Question 4Where two lines meet

The lines $-5x + 6y = -14$ and $-x + 4y = -14$ meet at the point $P$ . Find the coordinates of $P$ .

Show the worked solution

Step 1: P is on both lines, so its x and y fit both equations: solve them together. Number the equations $-5x + 6y = -14$ ... (1) $-x + 4y = -14$ ... (2)
Step 2: Make x the subject of (2): move the y-term across, then change every sign (divide by −1) $x = 4y + 14$ ... (3)
Step 3: Substitute (3) into (1): put the whole expression for x in a BRACKET $-5(4y + 14) + 6y = -14$
Step 4: Expand the bracket (the number in front multiplies BOTH terms), then collect $-20y - 70 + 6y = -14$ $-14y = 56$ $y = \dfrac{56}{-14}$ $= -4$
Step 5: Put your value of y into (3) to find x $x = 4(-4) + 14$ $= -2$ Check in (1): $-5(-2) + 6(-4)$ $= -14$ , the right-hand side of (1).