Question

Solve the system of equations by elimination

Answer 1

x = 4.5 , y = 1

Explanation:

given

2 -
`(y)/(2)` =
`(x)/(3)`

multiply through by 6 ( the LCM of 2 and 3 ) to clear the fractions

12 - 3y = 2x ( add 3y to both sides )

12 = 2x + 3y , or

2x + 3y = 12 → (1)

and

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

multiply both sides by 3 to clear the fraction

2(2x - 3y) = 12 ( divide both sides by 2 )

2x - 3y = 6 → (2)

We now have the 2 simplified equations to solve by elimination

2x + 3y = 12 → (1)

2x - 3y = 6 → (2)

add (1) and (2) term by term to eliminate y

(2x + 2x) + (3y - 3y) = 12 + 6

4x + 0 = 18

4x = 18 ( divide both sides by 4 )

x = 4.5

substitute x = 4.5 into either of the 2 equations and solve for y

substituting into (1)

2(4.5) + 3y = 12

9 + 3y = 12 ( subtract 9 from both sides )

3y = 3 ( divide both sides by 3 )

y = 1

solution is x = 4.5 , y = 1

Answer 2

`\sf x = (9)/(2)`

`\sf y = 1`

Explanation:

Let's solve the system of equations by elimination.

Given system:

`\sf 2 - (y)/(2) = (x)/(3)`

`\sf (2)/(3)(2x - 3y) = 4`

First, let's simplify equation (1) by multiplying both sides by 6 to eliminate the denominators:

`\sf 6 \cdot \left(2 - (y)/(2)\right) = 6 \cdot (x)/(3)`

This gives:

`\sf 12 - 3y = 2x`

Now, equation (2) can be simplified:

`\sf 4(2x - 3y) = 12`

`\sf 8x - 12y = 12`

Now, we have the system:

1.
`\sf 12 - 3y = 2x`

2.
`\sf 8x - 12y = 12`

To eliminate
`\sf x`, we can multiply equation (1) by 4:

`\sf 4(12 - 3y) = 4(2x)`

This simplifies to:

`\sf 48 - 12y = 8x`

Now, we have the system:

1.
`\sf 48 - 12y = 8x`

2.
`\sf 8x - 12y = 12`

Now, set the two expressions for
`\sf 8x` equal to each other:

`\sf 48 - 12y = 8x = 8x - 12y = 12`

Combine like terms:

`\sf -12y = -12`

Divide by -12:

`\sf y = 1`

Now substitute
`\sf y = 1` into one of the original equations. Let's use equation (1):

`\sf 12 - 3(1) = 2x`

`\sf 9 = 2x`

Divide by 2:

`\sf x = (9)/(2)`

So, the solution to the system is
`\sf x = (9)/(2)` and
`\sf y = 1`.