Question

What is the solution to the following system: y=6x+5 and 5x-4y=-1

Answer 1

x= -1 and y= -1

Explanation:

Power through with me as I explain this, its a bit long of an explanation.

To solve the following system of equations, we need to write 5x-4y=-1 in slope-intercept form.

`5x -4y= -1`

Subtract 5x from both sides.

`5x-5x -4y= -1 -5x`

`-4y= -1-5x`

Divide each term by -4.

`(-4y)/(-4) = (-1)/(-4) + (-5x)/(-4)`

Remember that dividing two negative values results in a positive value.

`y= (1)/(4) + (5x)/(4)`

Reorder the terms. (Reordering terms makes the work tidier, it does not change the result)

`y= (5)/(4) x+ (1)/(4)`

Now that we have the slope-intercept form, we can solve by subsitution with the two equations to find the solution.

Substitute
`(5)/(4) x + (1)/(4)` for y in
`y= 6x+5`.

`(5)/(4) x+(1)/(4) = 6x + 5`

Subtract 1/4 from each side.

`(5)/(4)x+(1)/(4)-(1)/(4)=6x+5-(1)/(4)`

Simplify the left side.

`(5)/(4)x=6x+5 -(1)/(4)`

Simplify the right side.

`(5)/(4)x=6x+(19)/(4)`

Subtract 6x from both sides.

`(5)/(4)x-6x=6x+(19)/(4)-6x`

Simplify.

`(5)/(4)x-6x=(19)/(4)`

Simplify the left side of the equation by factoring out x.

`x((5)/(4) -6)= (19)/(4)`

`x(-(19)/(4) )=(19)/(4)`

`-(19)/(4) x=(19)/(4)`

Multiply each side by 4.

`4\left(-(19)/(4)x\right)=(19*4)/(4)`

`-19x=19`

Divide both side by 19.

`(-19x)/(-19)=(19)/(-19)`

`x= -1`

Now that we know the value of x, we need to find y.

Insert the value of x in the equation y= 6x+5

`y= 6(-1) +5`

`y=-6+5`

`y=-1`

Thus, x= -1 and y= -1 OR (-1,-1)

Answer 2

(-1,-1)

Explanation:

Solve for the first variable in one of the equations, then substitute the result into the other equation.