Question

Solve the system of equations.

\begin{aligned} &-5x-3y - 9=0 \\\\ &4x-18y-54=0 \end{aligned}
​

−5x−3y−9=0
4x−18y−54=0
​

Answer 1

(0, - 3 )

Explanation:

- 5x - 3y - 9 = 0 → (1)

4x - 18y - 54 = 0 → (2)

multiplying (1) by - 6 and adding to (2) will eliminate y

30x + 18y + 54 = 0 → (3)

add (2) and (3) term by term to eliminate y

34x + 0 + 0 = 0

34x = 0 ⇒ x = 0

substitute x = 0 into either of the 2 equations and solve for y

substituting into (2)

4(0) - 18y - 54 = 0

- 18y - 54 = 0 ( add 54 to both sides )

- 18y = 54 ( divide both sides by - 18 )

y = - 3

solution is (0, - 3 )

Answer 2

(0, -3)

Explanation:

This system of equations consists of two equations. There are 3 main ways to solve a system of equations:

• Graphing (The solution is the point where the two lines intersect)
• Substitution
• Elimination

First, start by having the variables on one side.

`-5x-3y-9=0 \Rightarrow \text{Add 9 to both sides} \Rightarrow -5x-3y=9\\4x-18y-54=0 \Rightarrow \text{Add 54 to both sides} \Rightarrow 4x-18y=54 \Rightarrow \text{Simplify} \Rightarrow 2x-9y=27`

Solve Using Elimination

This method is the easiest to use in this situation.

In this method, we increase equations by a certain factor in order to eliminate one variable.

We can see that 3y in the first equation can be multiplied by 6 in order to obtain the 18y in the second equation. Therefore, we can multiply the whole first equation by 6:

`-30x-18y=54\\4x-18y=54`

Now, subtract the two equations to eliminate y.

`-34x=0\\x=0`

Plug in 0 to x in either of the equations to solve for y:

`-5(0)-3y=9\\0-3y=9\\-3y=9\\ \text{Divide both sides by -3}\\y=-3`

OR

`4(0)-18y=54\\0-18y=54\\-18y=54\\\text{Divide both sides by -18}\\y=-3`

Therefore:

(x, y) = (0, -3)