Question 218389


Start with the given system of equations:

{{{system(5x-2y=-1,7x+4y=53)}}}



{{{2(5x-2y)=2(-1)}}} Multiply the both sides of the first equation by 2.



{{{10x-4y=-2}}} Distribute and multiply.



So we have the new system of equations:

{{{system(10x-4y=-2,7x+4y=53)}}}



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



{{{(10x-4y)+(7x+4y)=(-2)+(53)}}}



{{{(10x+7x)+(-4y+4y)=-2+53}}} Group like terms.



{{{17x+0y=51}}} Combine like terms. Notice how the y terms cancel out.



{{{17x=51}}} Simplify.



{{{x=(51)/(17)}}} Divide both sides by {{{17}}} to isolate {{{x}}}.



{{{x=3}}} Reduce.



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{{{10x-4y=-2}}} Now go back to the first equation.



{{{10(3)-4y=-2}}} Plug in {{{x=3}}}.



{{{30-4y=-2}}} Multiply.



{{{-4y=-2-30}}} Subtract {{{30}}} from both sides.



{{{-4y=-32}}} Combine like terms on the right side.



{{{y=(-32)/(-4)}}} Divide both sides by {{{-4}}} to isolate {{{y}}}.



{{{y=8}}} Reduce.



So our answer is {{{x=3}}} and {{{y=8}}}.



Which form the ordered pair *[Tex \LARGE \left(3,8\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(3,8\right)]. So this visually verifies our answer.



{{{drawing(500,500,-7,13,-2,18,
grid(1),
graph(500,500,-7,13,-2,18,(-1-5x)/(-2),(53-7x)/(4)),
circle(3,8,0.05),
circle(3,8,0.08),
circle(3,8,0.10)
)}}} Graph of {{{5x-2y=-1}}} (red) and {{{7x+4y=53}}} (green)