Question

A system of equations consists of the two equations shown.

{4x+5y=18
6x−5y=20
Which procedure will produce a single equation in one variable? Select all the procedures that apply.
A. Subtract the first equation from the second equation.
B. Subtract the second equation from the first equation.
C. Multiply the first equation by 18; multiply the second equation by 18; add the equations.
D. Multiply the first equation by − 6; multiply the second equation by 4; add the two equations.
E. Multiply the first equation by 3; multiply the second equation by − 2; add the two equations.
F. Multiply the first equation by 3; multiply the second equation by 2; subtract the equations in any order.

Answer 1

C, D, E and F

Explanation:

Given

4x+5y=18

6x−5y=20

Required

Determine which procedure will result in a single equation in one variable

To do this; we'll test each of the options

A. Subtract the first equation from the second equation.

`(6x - 5y=20) - (4x+5y=18)`

`6x - 4x - 5y - 5y = 20 - 18`

`2x - 10y = 2` --- This didn't produce the desired result

B. Subtract the second equation from the first equation.

`(4x+5y=18) - (6x - 5y=20)`

`4x - 6x + 5y + 5y =18 - 20`

`-2x + 10y = -2` --- This didn't produce the desired result

C. Multiply the first equation by 18; multiply the second equation by 18; add the equations.

First Equation

`18 * (4x+5y=18)`

`72x + 90y = 324`

Second Equation

`18 * (6x - 5y=20)`

`108x - 90y = 360`

Add Resulting Equations

`(72x + 90y = 324) + (108x - 90y = 360)`

`72x + 108x + 90y - 90y = 324 + 360`

`72x + 108x = 324 + 360`

`180x = 684` --- This procedure is valid

D. Multiply the first equation by − 6; multiply the second equation by 4; add the two equations.

First Equation

`-6 * (4x+5y=18)`

`-24x - 30y = -108`

Second Equation

`4 * (6x - 5y=20)`

`24x - 20y = 80`

Add Resulting Equations

`(-24x - 30y = -108) + (24x - 20y = 80)`

`-24x + 24x - 30y -20y = -108+ 80`

`-50y = -28`

`50y = 28` --- This procedure is valid

E. Multiply the first equation by 3; multiply the second equation by − 2; add the two equations.

First Equation

`3 * (4x+5y=18)`

`12x + 15y = 54`

Second Equation

`-2 * (6x - 5y=20)`

`-12x + 10y = -40`

Add Resulting Equations

`(12x + 15y = 54) + (-12x + 10y = -40)`

`12x - 12x + 15y - 10y =54 - 40`

`5y = 14` --- This procedure is valid

F. Multiply the first equation by 3; multiply the second equation by 2; subtract the equations in any order

First Equation

`3 * (4x+5y=18)`

`12x + 15y = 54`

Second Equation

`2 * (6x - 5y=20)`

`12x - 10y = 40`

Subtract equation 1 from 2 or 2 from 1 will eliminate x;

Hence, the procedure is also valid;