**Solving a linear system by substitution :**

We can use substitution method to solve systems of linear equations by solving an equation for one variable and then substituting the resulting expression for that variable into the other equation.

**Step 1 : **

Solve one of the equations for one of its variables.

**Step 2 : **

Substitute the expression from step 1 into the other equation and solve for the other variable.

**Step 3 :**

Substitute the value from step 2 into either original equation and solve to find the value of the variable in step 1.

**Example 1 :**

Solve the system of linear equations by substitution. Check your answer by graphing.

4x + y = 8

-3x + y = 1

**Solution : **

**Step 1 :**

Solve an equation for one variable.

Select one of the equation, say -3x + y = 1.

Solve for the variable y in terms of x.

Add 3x on both sides.

(-3x + y) + 3x = (1) + 3x

-3x + y + 3x = 1 + 3x

Simplify.

y = 1 + 3x

**Step 2 : **

Substitute the expression for y in the other equation and solve.

4x + **y** = 8

4x + **(1 + 3x) ** = 8

Combine like terms.

7x + 1 = 8

Subtract 1 from both sides.

7x = 7

Divide both sides by 7.

7x / 7 = 7 / 7

x = 1

**Step 3 : **

Substitute the value of x we got above (x = 1) into one of the equations and solve for the other variable, y.

4x + y = 8

4(1) + y = 8

4 + y = 8

Subtract 4 from both sides.

y = 8

Hence, the solution of the system is (1, 8).

**Step 4 : **

Check the solution by graphing.

To graph the equations, write them in slope-intercept form.

That is,

y = mx + b

4x + y = 8

y = - 4x + 8

Slope = - 4

y-intercept = 8

-3x + y = 1

y = 3x + 1

Slope = 3

Y-intercept = 1

The point of intersection is (1, 4).

**Example 2 :**

Solve the system of linear equations by substitution. Check your answer by graphing.

x + y = 8

2x + y = 11

**Solution : **

**Step 1 :**

Solve an equation for one variable.

Select one of the equation, say x + y = 8.

Solve for the variable y in terms of x.

Subtract x from both sides.

(x + y) - x = (8) - x

x + y - x = 8 - x

Simplify.

y = 8 - x

**Step 2 : **

Substitute the expression for y in the other equation and solve.

2x + **y** = 11

2x + **(8 - x) ** = 11

Combine like terms.

x + 8 = 11

Subtract 8 from both sides.

x = 3

**Step 3 : **

Substitute the value of x we got above (x = 3) into one of the equations and solve for the other variable, y.

x + y = 8

3 + y = 8

Subtract 3 from both sides.

y = 5

Hence, the solution of the system is (3, 5).

**Step 4 : **

Check the solution by graphing.

To graph the equations, write them in slope-intercept form.

That is,

y = mx + b

x + y = 8

y = - x + 8

Slope = - 1

y-intercept = 8

2x + y = 11

y = -2x + 11

Slope = -2

Y-intercept = 11

The point of intersection is (1, 4).

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