We can use one of the following methods to solve a system of linear equations.

3. Cross Multiplication Method

**Example 1 :**

Solve by elimination method.

3x + 4y = 7

x - 4y = -3

**Solution :**

3x + 4y = 7 -----(1)

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

In the given two equations, y-term has the same coefficient and different signs. By adding the above two equations, we can eliminate y-term and solve for y.

(1) + (2) :

4x = 4

Divide each side by 4.

x = 1

Substitute x = 1 in (1).

3(1) + 4y = 7

3 + 4y = 7

Subtract 3 from each side.

4y = 4

Divide each side by 4.

y = 1

So, the solution is

(x, y) = (1, 1)

**Example 2 :**

Solve by elimination method.

3x + 4y = -25

2x - 3y = 6

**Solution :**

3x + 4y = -25 -----(1)

2x - 3y = 6 -----(2)

Both x terms and y terms have different coefficients in the above system of equations.

Let's try to make the coefficients of y terms equal.

To make the coefficients of y terms equal, we have to find the least common multiple 4 and 3.

The least common multiple of 4 and 3 is 12.

Multiply the first equation by 3 in order to make the coefficient of y as 12 and multiply the second equation by 4 in order to make the coefficient of y as -12.

(1) ⋅ 3 ----> 9x + 12y = -75

(2) ⋅ 4 ----> 8x - 12y = 24

Now, we can add the two equations and eliminate y as shown below.

Divide each side by 17.

x = -3

Substitute -3 for x in (1).

(1)----> 3(-3) + 4y = -25

-9 + 4y = -25

Add 9 to each side.

4y = -16

Divide each side by 4.

y = -4

So, the solution is

(x, y) = (-3, -4)

**Example 3 : **

Solve for x and y using substitution.

x - 5y + 17 = 0

2x + y + 1 = 0

**Solution : **

x - 5y + 17 = 0 -----(1)

2x + y + 1 = 0 -----(2)

**Step 1 :**

Solve (1) for x.

x - 5y + 17 = 0

Subtract 17 from each side.

x - 5y = -17

Add 5y to each side.

x = 5y - 17 -----(3)

**Step 2 : **

Substitute (5y - 17) for x into (2).

(2)-----> 2(5y - 17) + y + 1 = 0

10y - 34 + y + 1 = 0

11y - 33 = 0

Add 33 to each side.

11y = 33

Divide each side by 11.

y = 3

**Step 3 :**

Substitute 3 for y into (3).

(3)-----> x = 5(3) - 17

x = 15 - 17

x = -2

So, the solution is

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

**Example 4 :**

Solve the following system of equations using cross multiplication method.

2x + 7y - 5 = 0

-3x + 8y = -11

**Solution:**

First we have to change the given linear equations in the form a_{1}x + b_{1}y + c_{1} = 0, a_{2}x + b_{2}y + c_{2} = 0.

2x + 7y - 5 = 0

-3x + 8y + 11 = 0

x/(77 + 40) = y/(15 - 22) = 1/[16 + 21]

x/117 = y/(-7) = 1/37

x/117 = 1/37 x = 117/37 |
y/(-7) = 1/37 y = -7/37 |

So, the solution is

(x, y) = (117/37, -7/37)

**Example 5 :**

Solve the following system of equations by graphing.

x + y - 4 = 0

3x - y = 0

**Solution :**

**Step 1 :**

Let us re-write the given equations in slope-intercept form (y = mx + b).

y = - x + 4

(slope is -1 and y-intercept is 4)

y = 3x

(slope is 3 and y-intercept is 0)

Based on slope and y-intercept, we can graph the given equations.

**Step 2 :**

Find the point of intersection of the two lines. It appears to be (1, 3). Substitute to check if it is a solution of both equations.

x + y - 4 = 0 1 + 3 - 4 = 0 4 - 4 = 0 0 = 0 |
3x - y = 0 3(1) - 3 = 0 3 - 3 = 0 0 = 0 |

Because the point (1, 3) satisfies both the equations, the solution for the given system is (1, 3).

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