How do solve the following linear system?: `5x+y=6 , 12x-2y=-16`?

Solve the linear system:

`"Equation 1":` `5x+y=6`

`"Equation 2":` `12x-2y=-16`

We can solve the system by elimination.

Multiply Equation 1 by `2`.

`2(5x+y=6)`

`10x+2y=12`

Add Equation 1 and Equation 2.

`10x+2y=color(white)(....)12`
`12x-2y=-16`
`-------`
`22xcolor(white)(........)=-4`

Divide both sides by `22`.

`x=-4/22`

Simplify.

`x=-2/11` or `~~-0.18`

Substitute the value for `x` into Equation 1. Solve for `y`.

`5x+y=6`

`5(-2/11)+y=6`

Expand.

`-10/11+y=6`

Add `10/11` to both sides.

`y=6+10/11`

Multiply `6` by `11/11` to get an equivalent fraction with `11` as the denominator.

`y=6xx11/11+10/11`

`y=66/11+10/11`

`y=76/11` or `~~6.9`

The solution to the linear system is `(-2/11,76/11)`.

The approximate solution is `(-0.18,6.9)`.

graph{(5x+y-6)(12x-2y+16)=0 [-7.52, 6.53, 1.603, 8.627]}

Given -

`5x+y=6` --------- (1)
`12x-2y=-16` ------(2)

`5x+y=6` --------- (1) `xx 2`
`12x-2y=-16` ------(2)

`10x+2y=12`--------(3)
`12x-2y=-16` -------(2) -- `(3)+(2)`
`22x=-4`
`x=-4/22=-2/11`

`x=-2/11`

Plug in `x=-2/11` in equation (1)

`5(-2/11)+y=6`
`-10/11+y=6`

`y=6+10/11=(66+10)/11=76/11`

`y=76/11`

Many ways, but my favorite is the elimination method, and fortunately, it works in this situation!

Let's make the equations look neater (put in `y=mx+b` form)

• Equation 1: `" "5x+y=6 or y=-5x+6`
• Equation 2: `" "12x-2y=-16 or y=6x+8`

The elimination method allows us to cancel out one of the variables, making it an easy algebra equation to solve for the remaining variable. You'll see.

Let's eliminate the `7` variable. In order to do that, we need to simply multiply one of the equations by negative one, let's do it to equation one.

• Equation 1: `" "-y=+5x-6`
• Equation 2: `" " y=6x+8`

Now, we add the equations together, getting one combined equation:

`0=11x+2`

`x=-2/11`

Now, we plug in the `x` value into one of the ORIGINAL equations to solve for `y`.

Equaiton 1:

`y=-5(-2/11)+6`

`y=76/11`