**Solving Systems of Equations by Substitution :**

In this section, you will learn how to solve system of linear equations with two unknowns using substitution.

**Step 1 : **

In the given two equations, solve one of the equations either for x or y.

**Step 2 : **

Substitute the result of step 1 into other equation and solve for the second variable.

**Step 3 : **

Using the result of step 2 and step 1, solve for the first variable.

**Example 1 :**

Solve the following system of equations by substitution.

x - 5y + 17 = 0 and 2x + y + 1 = 0

**Solution :**

The above explained steps have been illustrated in the picture shown below.

Therefore, the solution is

(x, y) = (-2, 3)

**Example 2 :**

Solve the following system of equations by substitution.

5x - 3y - 8 = 0 and 2x - 3y - 5 = 0

**Solution :**

5x - 3y - 8 = 0 -----(1)

2x - 3y - 5 = 0 -----(2)

**Step 1 :**

In both (1) and (2), we have the same coefficient for y.

So, solve (1) for 3y.

5x - 3y - 8 = 0

-3y = 8 - 5x

Multiply each side by (-1).

3y = 5x - 8 -----(3)

**Step 2 : **

Substitute (5x - 8) for y into (2).

(2)-----> 2x - (5x - 8) - 5 = 0

2x - 5x + 8 - 5 = 0

-3x + 3 = 0

Subtract 3 from each side.

-3x = -3

Divide each side by (-3).

x = 1

**Step 3 :**

Substitute 1 for x into (3).

(3)-----> 3y = 5(1) - 8

3y = -3

Divide each side by 3.

y = -1

Therefore, the solution is

(x, y) = (1, -1)

**Example 3 :**

Solve the following system of equations by substitution.

y = 6x - 11 and -2x - 3y = -7

**Solution :**

y = 6x - 11 -----(1)

-2x - 3y = -7 -----(2)

**Step 1 :**

In the given two equations, already (1) is solved for y.

So, we don't have to do anything more in this step.

**Step 2 : **

Substitute (6x - 11) for y into (2).

(2)-----> -2x - 3(6x - 11) = -7

-2x - 18x + 33 = -7

-20x + 33 = -7

Subtract 33 from each side.

-20x = -40

Divide each side by (-20).

x = 2

**Step 3 :**

Substitute 2 for x into (1).

(1)-----> y = 6(2) - 11

y = 12 - 11

y = 1

Therefore, the solution is

(x, y) = (2, 1)

**Example 4 :**

Solve the following system of equations by substitution.

2x - 3y = -1 and y = x - 1

**Solution :**

2x - 3y = -1 -----(1)

y = x - 1 -----(2)

**Step 1 :**

In the given two equations, already (2) is solved for y.

So, we don't have to do anything more in this step.

**Step 2 : **

Substitute (x - 1) for y into (1).

(2)-----> 2x - 3(x - 1) = -1

2x - 3x + 3 = -1

-x + 3 = -1

Subtract 3 from each side.

-x = -4

Multiply each side by (-1).

x = 4

**Step 3 :**

Substitute 4 for x into (2).

(2)-----> y = 4 - 1

y = 3

Therefore, the solution is

(x, y) = (4, 3)

**Example 5 :**

Solve the following equations by substitution method

y = -3x + 5 and 5x - 4y = -3

**Solution :**

y = -3x + 5 -----(1)

5x - 4y = -3 -----(2)

**Step 1 :**

In the given two equations, already (1) is solved for y.

So, we don't have to do anything more in this step.

**Step 2 : **

Substitute (-3x + 5) for y into (2).

(2)-----> 5x - 4(-3x + 5) = -3

5x + 12x - 20 = -3

17x - 20 = -3

Add 20 to each side.

17x = 17

Divide each side by 17.

x = 1

**Step 3 :**

Substitute 1 for x into (1).

(1)-----> y = -3(1) + 5

y = -3 + 5

y = 2

Therefore, the solution is

(x, y) = (1, 2)

After having gone through the stuff given above, we hope that the students would have understood how to solve system of equations by substitution.

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

You can also visit the following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**