# System of Linear Equations

See how to solve a system of linear equations

and graph a system of linear inequalities.

16 examples and their solutions.

- x - y = 4

2x + y = 5

→ Using Graph - Substitution Method:

x = y + 4

2x + y = 5

→ 2(y + 4) + y = 5 - Elimination Method:

x - y = 4

+2x + y = 5

3x = 9 - x + y > 2

2x - y ≥ 3 - x ≥ 0

y ≥ 0

3x + y ≤ 9

x + 2y ≤ 8

→ Maximum value of x + y? - x + y + z = 3

2x - y + z = 6

x + 2y - z = -4

## System of Linear Equations: Using Graph

### Example

x - y = 4

2x + y = 5

Solution 2x + y = 5

x - y = 4

-y = -x + 4

y = x - 4- [1]

2x + y = 5

y = -2x + 5- [1]

[2]

(3, 1)

[1] To graph the linear equations,

change the linear equations to [y = ...].

[2] Graph y = x - 4 and y = -2x + 5.

The intersecting point is (3, 1).

Close

-y = -x + 4

y = x - 4- [1]

2x + y = 5

y = -2x + 5- [1]

(3, 1)

[1] To graph the linear equations,

change the linear equations to [y = ...].

[2] Graph y = x - 4 and y = -2x + 5.

The intersecting point is (3, 1).

Close

### Example

x - y = 4

2x - 2y = 8

Solution 2x - 2y = 8

x - y = 4

-y = -x + 4

y = x - 4

2x - 2y = 8

x - y = 4

-y = -x + 4

y = x - 4

Infinitely many solutions- [1]

[1] x - y = 4 → y = x - 4

2x - 2y = 8 → y = x - 4

Both graphs are the same.

So there are infinitely many intersecting points.

So this system has

infinitely many solutions.

Close

-y = -x + 4

y = x - 4

2x - 2y = 8

x - y = 4

-y = -x + 4

y = x - 4

Infinitely many solutions- [1]

[1] x - y = 4 → y = x - 4

2x - 2y = 8 → y = x - 4

Both graphs are the same.

So there are infinitely many intersecting points.

So this system has

infinitely many solutions.

Close

### Example

x - y = 4

x - y = -3

Solution x - y = -3

x - y = 4

-y = -x + 4

y = x - 4

x - y = -3

-y = -x - 3

y = x + 3

No solution- [1]

[1] x - y = 4 → y = x - 4

x - y = -3 → y = x + 3

These two lines are parallel.

So these two lines don't meet.

(= no intersecting points)

So this system has

no solution.

Close

-y = -x + 4

y = x - 4

x - y = -3

-y = -x - 3

y = x + 3

No solution- [1]

[1] x - y = 4 → y = x - 4

x - y = -3 → y = x + 3

These two lines are parallel.

So these two lines don't meet.

(= no intersecting points)

So this system has

no solution.

Close

## Substitution Method

### Example

x - y = 4

2x + y = 5

Solution 2x + y = 5

x - y = 4

x = y + 4

2(y + 4) + y = 5- [1]

2y + 8 + y = 5

3y = -3

y = -1

x = -1 + 4- [2]

= 3

x = 3, y = -1

[1] x = y + 4 → 2x + y = 5

[2] y = -1 → x = y + 4

Close

x = y + 4

2(y + 4) + y = 5- [1]

2y + 8 + y = 5

3y = -3

y = -1

x = -1 + 4- [2]

= 3

x = 3, y = -1

[1] x = y + 4 → 2x + y = 5

[2] y = -1 → x = y + 4

Close

### Example

x - y = 4

2x - 2y = 8

Solution 2x - 2y = 8

x - y = 4

x = y + 4

2(y + 4) - 2y = 8- [1]

2y + 8 - 2y = 8

0 = 0

Infinitely many solutions- [2]

[1] x = y + 4 → 2x - 2y = 8

[2] 0 = 0

This is always true.

If you get an equation that is always true,

then the system has

infinitely many solutions.

Close

x = y + 4

2(y + 4) - 2y = 8- [1]

2y + 8 - 2y = 8

0 = 0

Infinitely many solutions- [2]

[1] x = y + 4 → 2x - 2y = 8

[2] 0 = 0

This is always true.

If you get an equation that is always true,

then the system has

infinitely many solutions.

Close

### Example

x - y = 4

x - y = -3

Solution x - y = -3

x - y = 4

x = y + 4

y + 4 - y = -3- [1]

4 = -3

No solution- [2]

[1] x = y + 4 → x - y = -3

[2] 4 = -3

This is always false.

If you get an equation that is always false,

then the system has

no solution.

Close

x = y + 4

y + 4 - y = -3- [1]

4 = -3

No solution- [2]

[1] x = y + 4 → x - y = -3

[2] 4 = -3

This is always false.

If you get an equation that is always false,

then the system has

no solution.

Close

## Elimination Method

### Example

x - y = 4

2x + y = 5

Solution 2x + y = 5

x - y = 4

+2x + y = 5

3x = 9- [1]

x = 3

3 - y = 4- [2]

-y = 1

y = -1

x = 3, y = -1

[1] To remove -y and +y,

add the equations.

[2] x = 3 → x - y = 4

Close

+2x + y = 5

3x = 9- [1]

x = 3

3 - y = 4- [2]

-y = 1

y = -1

x = 3, y = -1

[1] To remove -y and +y,

add the equations.

[2] x = 3 → x - y = 4

Close

### Example

3x + 2y = 7

2x - 3y = -4

Solution 2x - 3y = -4

6x + 4y = 14

-6x - 9y = -12- [1]

13y = 26- [2]

y = 2

3x + 2⋅2 = 7- [3]

3x + 4 = 7

3x = 3

x = 1

x = 1, y = 2

[1] First make 3x and 2x

the same (or the opposites).

3x + 2y = 7 [×2]→ 6x + 4y = 14

2x - 3y = -4 [×3]→ 6x - 9y = -12

[2] To remove 6x and 6x,

subtract the equations.

[3] y = 2 → 3x + 2y = 7

Close

-6x - 9y = -12- [1]

13y = 26- [2]

y = 2

3x + 2⋅2 = 7- [3]

3x + 4 = 7

3x = 3

x = 1

x = 1, y = 2

[1] First make 3x and 2x

the same (or the opposites).

3x + 2y = 7 [×2]→ 6x + 4y = 14

2x - 3y = -4 [×3]→ 6x - 9y = -12

[2] To remove 6x and 6x,

subtract the equations.

[3] y = 2 → 3x + 2y = 7

Close

### Example

x - y = 4

2x - 2y = 8

Solution 2x - 2y = 8

2x - 2y = 8- [1]

-2x - 2y = 8

0 = 0- [2]

Infinitely many solutions- [3]

[1] First make x and 2x

the same (or the opposites).

x - y = 4 [×2]→ 2x - 2y = 8

2x - 2y = 8

[2] To remove 2x and 2x,

subtract the equations.

[3] 0 = 0

This is always true.

If you get an equation that is always true,

then the system has

infinitely many solutions.

Close

-2x - 2y = 8

0 = 0- [2]

Infinitely many solutions- [3]

[1] First make x and 2x

the same (or the opposites).

x - y = 4 [×2]→ 2x - 2y = 8

2x - 2y = 8

[2] To remove 2x and 2x,

subtract the equations.

[3] 0 = 0

This is always true.

If you get an equation that is always true,

then the system has

infinitely many solutions.

Close

### Example

x - y = 4

x - y = -3

Solution x - y = -3

x - y = 4

-x - y = -3

0 = 7- [1]

No solution- [2]

[1] To remove x and x,

subtract the equations.

[2] 0 = 7

This is always false.

If you get an equation that is always false,

then the system has

no solution.

Close

-x - y = -3

0 = 7- [1]

No solution- [2]

[1] To remove x and x,

subtract the equations.

[2] 0 = 7

This is always false.

If you get an equation that is always false,

then the system has

no solution.

Close

## System of Linear Inequalities

### Example

x + y > 2

2x - y ≥ 3

Solution 2x - y ≥ 3

x + y > 2

y > -x + 2

2x - y ≥ 3

-y ≥ -2x + 3

y ≤ 2x - 3- [1]

[2]

[1] To graph the linear inequalities,

change the linear inequalities to [y = ...].

[2] Graph y > -x + 2 and y ≤ 2x - 3.

Color the intersecting region.

The intersecting region is the answer.

Close

y > -x + 2

2x - y ≥ 3

-y ≥ -2x + 3

y ≤ 2x - 3- [1]

[1] To graph the linear inequalities,

change the linear inequalities to [y = ...].

[2] Graph y > -x + 2 and y ≤ 2x - 3.

Color the intersecting region.

The intersecting region is the answer.

Close

### Example

2x - y > -3

2x - y ≥ 1

Solution 2x - y ≥ 1

2x - y > -3

-y > -2x - 3

y < 2x + 3

2x - y ≥ 1

-y ≥ -2x + 1

y ≤ 2x - 1

[1]

[1] The intersecting region is y ≤ 2x - 1.

Close

-y > -2x - 3

y < 2x + 3

2x - y ≥ 1

-y ≥ -2x + 1

y ≤ 2x - 1

[1] The intersecting region is y ≤ 2x - 1.

Close

### Example

3x + y ≥ 4

3x + y < -2

Solution 3x + y < -2

3x + y ≥ 4

y ≥ -3x + 4

3x + y < -2

y < -3x -2

No solution- [1]

[1] Graph y ≥ -3x + 4 and y < -3x -2.

There's no the intersecting region.

So

no solution

is the answer.

Close

y ≥ -3x + 4

3x + y < -2

y < -3x -2

No solution- [1]

[1] Graph y ≥ -3x + 4 and y < -3x -2.

There's no the intersecting region.

So

no solution

is the answer.

Close

## Linear Programming

### Example

For the given system of linear inequalities,

find the maximum value of x + y.

x ≥ 0

y ≥ 0

3x + y ≤ 9

x + 2y ≤ 8

Solution find the maximum value of x + y.

x ≥ 0

y ≥ 0

3x + y ≤ 9

x + 2y ≤ 8

x ≥ 0

y ≥ 0

3x + y ≤ 9

y ≤ -3x + 9

x + 2y ≤ 8

2y ≤ -x + 8

y ≤ -12x + 4- [1]

x + y = k

y = -x + k- [2]

- [3]

-3x + 9 = -12x + 4- [4]

-6x + 18 = -x + 8

-5x = -10

x = 2

y = -3⋅2 + 9- [5]

= -6 + 9

= 3

(x, y) = (2, 3)

k = 2 + 3- [6]

= 5

[1] To graph the linear inequalities,

change the linear inequalities to [y = ...].

[2] Set x + y = k.

→ y = -x + k

[3] Graph the linear inequalities

on a coordinate plane.

Color the intersecting region.

Draw y = -x + k.

k is the y-intercept of the line.

So k is maximum

when the line passes through the green point.

(slope: -3 < -1 < -1/2)

[4] The green point is the intersecting point of

y = -3x + 9 and y = [-1/2]x + 4.

So, to find the green point,

set -3x + 9 = [-1/2]x + 4.

[5] x = 2 → y = -3x + 9

[6] The green point is (2, 3).

→ k = x + y

(x + y = k)

Close

y ≥ 0

3x + y ≤ 9

y ≤ -3x + 9

x + 2y ≤ 8

2y ≤ -x + 8

y ≤ -12x + 4- [1]

x + y = k

y = -x + k- [2]

-3x + 9 = -12x + 4- [4]

-6x + 18 = -x + 8

-5x = -10

x = 2

y = -3⋅2 + 9- [5]

= -6 + 9

= 3

(x, y) = (2, 3)

k = 2 + 3- [6]

= 5

[1] To graph the linear inequalities,

change the linear inequalities to [y = ...].

[2] Set x + y = k.

→ y = -x + k

[3] Graph the linear inequalities

on a coordinate plane.

Color the intersecting region.

Draw y = -x + k.

k is the y-intercept of the line.

So k is maximum

when the line passes through the green point.

(slope: -3 < -1 < -1/2)

[4] The green point is the intersecting point of

y = -3x + 9 and y = [-1/2]x + 4.

So, to find the green point,

set -3x + 9 = [-1/2]x + 4.

[5] x = 2 → y = -3x + 9

[6] The green point is (2, 3).

→ k = x + y

(x + y = k)

Close

## System of Equations (Three Variables)

### Example

x + y + z = 3

2x - y + z = 6

x + 2y - z = -4

Solution 2x - y + z = 6

x + 2y - z = -4

2x - y + z = 6

-x + y + z = 3

x - 2y = 3- [1]

x + y + z = 3

+x + 2y - z = -4

2x + 3y = -1- [2]

2x - 4y = 6

-2x + 3y = -1- [3]

7y = -7- [4]

y = -1

x - 2⋅(-1) = 3- [5]

x + 2 = 3

x = 1

1 + (-1) + z = 3- [6]

z = 3

x = 1, y = -1, z = 3

[1] Pick 2x - y + z = 6 and x + y + z = 3.

To remove +z and +z,

subtract the equations.

[2] Pick x + y + z = 3 and x + 2y - z = -4.

To remove +z and -z,

add the equations.

[3] x - 2y = 3

2x + 3y = -1

Solve this system of linear equations.

First make x and 2x

the same (or the opposites).

x - 2y = 3 [×2]→ 2x - 6y = 9

2x + 3y = -1

[4] To remove 2x and 2x,

subtract the equations.

[5] y = -1 → x - 2y = 3

[6] x = 1, y = -1 → x + y + z = 3

Close

-x + y + z = 3

x - 2y = 3- [1]

x + y + z = 3

+x + 2y - z = -4

2x + 3y = -1- [2]

2x - 4y = 6

-2x + 3y = -1- [3]

7y = -7- [4]

y = -1

x - 2⋅(-1) = 3- [5]

x + 2 = 3

x = 1

1 + (-1) + z = 3- [6]

z = 3

x = 1, y = -1, z = 3

[1] Pick 2x - y + z = 6 and x + y + z = 3.

To remove +z and +z,

subtract the equations.

[2] Pick x + y + z = 3 and x + 2y - z = -4.

To remove +z and -z,

add the equations.

[3] x - 2y = 3

2x + 3y = -1

Solve this system of linear equations.

First make x and 2x

the same (or the opposites).

x - 2y = 3 [×2]→ 2x - 6y = 9

2x + 3y = -1

[4] To remove 2x and 2x,

subtract the equations.

[5] y = -1 → x - 2y = 3

[6] x = 1, y = -1 → x + y + z = 3

Close

### Example

x + 4y = 5

y + 4z = 7

z + 4x = 8

x + y + z = ?

Solution y + 4z = 7

z + 4x = 8

x + y + z = ?

x + 4y = 5

y + 4z = 7

+4x + z = 8

5x + 5y + 5z = 20

5(x + y + z) = 20

x + y + z = 4

y + 4z = 7

+4x + z = 8

5x + 5y + 5z = 20

5(x + y + z) = 20

x + y + z = 4