Solve a System of Equations

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Key Ideas Easy Medium Hard Super Challenge

Substitute · Eliminate · Graph

Check in both equations

What to Remember

Substitution

Solve one equation for a variable, then replace that variable in the other equation. After you find one value, substitute back to find the other.

Elimination

Add or subtract the equations so one variable cancels. Multiply first if the coefficients are not opposites.

Graphing

Write both equations as y = mx + b. Graph both lines. The intersection is the solution. Check it in both original equations.

Always check

Plug the ordered pair into both original equations. Both must be true.

Level 1

Easy

Substitution when y is already solved, elimination by adding, or graph two slope-intercept lines.

Example

Solve by substitution. y = x + 3x + y = 11

Substitute y = x + 3 into x + y = 11.

x + (x + 3) = 11

2x + 3 = 11

2x = 8

x = 4

y = 4 + 3 = 7

Check.

4 + 7 = 11

Answer: (4, 7)

Solve by substitution. y = 52x + y = 13

Show Answer

2x + 5 = 13

2x = 8

x = 4

y = 5

Check.

2(4) + 5 = 13

Answer: (4, 5)

Solve by elimination. x + y = 9x − y = 3

Show Answer

Add the equations.

2x = 12

x = 6

6 + y = 9

y = 3

Check.

6 − 3 = 3

Answer: (6, 3)

Solve by graphing. y = xy = −x + 6

Show Answer

y = x has slope 1 and y-intercept (0, 0).

y = −x + 6 has slope −1 and y-intercept (0, 6).

The lines meet at (3, 3).

Check.

3 = 3 and 3 = −3 + 6

Answer: (3, 3)

Solve by substitution. x = 2yx + 3y = 15

Show Answer

Substitute x = 2y into x + 3y = 15.

2y + 3y = 15

5y = 15

y = 3

x = 2(3) = 6

Check.

6 + 3(3) = 15

Answer: (6, 3)

Solve by elimination. 2x + y = 10x − y = 2

Show Answer

Add the equations.

3x = 12

x = 4

4 − y = 2

−y = −2

y = 2

Check.

2(4) + 2 = 10

Answer: (4, 2)

Level 2

Medium

Solve one equation for a variable first, or multiply one equation before you add.

Example

Solve. 2x + y = 9x − y = 0

Add the equations.

3x = 9

x = 3

3 − y = 0

y = 3

Check.

2(3) + 3 = 9

Answer: (3, 3)

Solve. 3x + 2y = 16x + 2y = 10

Show Answer

Subtract the second from the first.

2x = 6

x = 3

3 + 2y = 10

2y = 7

y = 72

Check.

3(3) + 272 = 9 + 7 = 16

Answer: (3, 72)

Solve. 2x + 3y = 11x − y = 3

Show Answer

Solve the second for x.

x = y + 3

Substitute into 2x + 3y = 11.

2(y + 3) + 3y = 11

2y + 6 + 3y = 11

5y + 6 = 11

5y = 5

y = 1

x = 1 + 3 = 4

Check.

2(4) + 3(1) = 11

Answer: (4, 1)

Solve. 2x + 3y = 124x − y = 5

Show Answer

Multiply the second equation by 3.

12x − 3y = 15

Add to 2x + 3y = 12.

14x = 27

x = 2714

42714 − y = 5

10814 − y = 5

547 − y = 357

− y = 357 − 547

− y = −197

y = 197

Check.

22714 + 3197 = 277 + 577 = 847 = 12

Answer: (2714, 197)

Solve by graphing. y = 2x − 1y = −x + 5

Show Answer

First line: slope 2, y-intercept (0, −1).

Second line: slope −1, y-intercept (0, 5).

The lines meet at (2, 3).

Check.

2(2) − 1 = 3 and −2 + 5 = 3

Answer: (2, 3)

Solve. 3x + y = 142x + y = 9

Show Answer

Subtract the second from the first.

x = 5

2(5) + y = 9

10 + y = 9

y = −1

Check.

3(5) + (−1) = 15 − 1 = 14

Answer: (5, −1)

Level 3

Hard

Multiply both equations, or expect a fractional solution. Graph after rewriting in slope-intercept form.

Example

Solve. 2x + 3y = 133x + 5y = 21

Multiply the first equation by 5 and the second by 3.

10x + 15y = 65

9x + 15y = 63

Subtract.

x = 2

2(2) + 3y = 13

4 + 3y = 13

3y = 9

y = 3

Check.

3(2) + 5(3) = 6 + 15 = 21

Answer: (2, 3)

Solve. 3x + 2y = 82x + 4y = 9

Show Answer

Multiply the first equation by 2.

6x + 4y = 16

Subtract 2x + 4y = 9.

4x = 7

x = 74

274 + 4y = 9

72 + 4y = 9

4y = 9 − 72

4y = 182 − 72

4y = 112

y = 112 × 14

y = 118

Check.

374 + 2118 = 214 + 114 = 324 = 8

Answer: (74, 118)

Solve. 2x − 3y = 54x + y = 11

Show Answer

Multiply the second equation by 3.

12x + 3y = 33

Add to 2x − 3y = 5.

14x = 38

x = 3814

x = 197

4197 + y = 11

767 + y = 777

y = 17

Check.

2197 − 317 = 387 − 37 = 357 = 5

Answer: (197, 17)

Solve by graphing. x + y = 6y = −2x + 3

Show Answer

Rewrite x + y = 6 as y = −x + 6.

Graph y = −x + 6 and y = −2x + 3.

The lines meet at (−3, 9).

Check.

−3 + 9 = 6

−2(−3) + 3 = 6 + 3 = 9

Answer: (−3, 9)

Solve. 5x − 2y = 93x + 4y = 1

Show Answer

Multiply the first equation by 2.

10x − 4y = 18

Add to 3x + 4y = 1.

13x = 19

x = 1913

31913 + 4y = 1

5713 + 4y = 1313

4y = 1313 − 5713

4y = −4413

y = −1113

Check.

51913 − 2−1113 = 9513 + 2213 = 11713 = 9

Answer: (1913, −1113)

Solve. y = 12x + 12x − 4y = −4

Show Answer

Substitute y = 12x + 1 into 2x − 4y = −4.

2x − 412x − 4(1) = −4

2x − 2x − 4 = −4

−4 = −4

The equation is always true, so the lines are the same.

Infinitely many solutions. Every point on y = 12x + 1.

Answer: infinitely many solutions

Super Challenge

Solve. x2 + y3 = 4x − y = 2

Show Answer

Multiply the first equation by 6 to clear the fractions.

3x + 2y = 24

From x − y = 2, x = y + 2.

3(y + 2) + 2y = 24

3y + 6 + 2y = 24

5y = 18

y = 185

x = 185 + 2 = 285

Check.

285 − 185 = 2

Answer: (285, 185)

The system 2x + ky = 8 and 4x + 6y = 16 has exactly one solution. What must be true of k, and what is that solution when k = 1?

Show Answer

Multiply the first equation by 2: 4x + 2ky = 16.

Compare with 4x + 6y = 16.

If 2k = 6, then k = 3 and the equations are the same: infinitely many solutions.

For exactly one solution, k ≠ 3.

When k = 1: 2x + y = 8 and 4x + 6y = 16.

Multiply the first by 2: 4x + 2y = 16.

Subtract from 4x + 6y = 16.

4y = 0, so y = 0

2x = 8, so x = 4

The solution is (4, 0).

Answer: k ≠ 3; when k = 1, (4, 0)

Solve by graphing, then confirm with substitution. 2x + 2y = 10y = −x + 8

Show Answer

2x + 2y = 10

x + y = 5

y = −x + 5

Set −x + 5 = −x + 8.

5 = 8

5 = 8 is false, so there is no intersection.

The lines are parallel: both have slope −1 and different intercepts 5 and 8.

Answer: no solution

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