Question 178815

{{{system(2x+5y=4,x+5y=7)}}}

{{{-1(x+5y)=-1(7)}}} Multiply the both sides of the second equation by -1.

{{{-1x-5y=-7}}} Distribute and multiply.

So we have the new system of equations:

{{{system(2x+5y=4,-1x-5y=-7)}}}

Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:

{{{(2x+5y)+(-1x-5y)=(4)+(-7)}}}

{{{(2x+-1x)+(5y+-5y)=4+-7}}} Group like terms.

{{{x+0y=-3}}} Combine like terms.

{{{x=-3}}} Simplify.

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{{{2x+5y=4}}} Now go back to the first equation.

{{{2(-3)+5y=4}}} Plug in {{{x=-3}}}.

{{{-6+5y=4}}} Multiply.

{{{5y=4+6}}} Add {{{6}}} to both sides.

{{{5y=10}}} Combine like terms on the right side.

{{{y=(10)/(5)}}} Divide both sides by {{{5}}} to isolate {{{y}}}.

{{{y=2}}} Reduce.

So our answer is {{{x=-3}}} and {{{y=2}}}.

Which form the ordered pair *[Tex \LARGE \left(-3,2\right)].

This means that the system is consistent and independent.

Notice when we graph the equations, we see that they intersect at *[Tex \LARGE \left(-3,2\right)]. So this visually verifies our answer.

{{{drawing(500,500,-13,7,-8,12,
grid(1),
graph(500,500,-13,7,-8,12,(4-2x)/(5),(7-x)/(5)),
circle(-3,2,0.05),
circle(-3,2,0.08),
circle(-3,2,0.10)
)}}} Graph of {{{2x+5y=4}}} (red) and {{{x+5y=7}}} (green)