Question 146544

Start with the given system of equations:

{{{system(-7x+6y=59,-8x+6y=58)}}}



{{{-1(-8x+6y)=-1(58)}}} Multiply the both sides of the second equation by -1.



{{{8x-6y=-58}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-7x+6y=59,8x-6y=-58)}}}



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



{{{(-7x+6y)+(8x-6y)=(59)+(-58)}}}



{{{(-7x+8x)+(6y+-6y)=59+-58}}} Group like terms.



{{{x+0y=1}}} Combine like terms. Notice how the y terms cancel out.



{{{x=1}}} Simplify.



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{{{-7x+6y=59}}} Now go back to the first equation.



{{{-7(1)+6y=59}}} Plug in {{{x=1}}}.



{{{-7+6y=59}}} Multiply.



{{{6y=59+7}}} Add {{{7}}} to both sides.



{{{6y=66}}} Combine like terms on the right side.



{{{y=(66)/(6)}}} Divide both sides by {{{6}}} to isolate {{{y}}}.



{{{y=11}}} Reduce.



So our answer is {{{x=1}}} and {{{y=11}}}.



Which form the ordered pair *[Tex \LARGE \left(1,11\right)].



This means that the two equations are consistent and independent.



Notice when we graph the equations, we see that they intersect at *[Tex \LARGE \left(1,11\right)]. So this visually verifies our answer.



{{{drawing(500,500,-9,11,-3,21,
grid(1),
graph(500,500,-9,11,-3,21,(59+7x)/(6),(58+8x)/(6)),
circle(1,11,0.05),
circle(1,11,0.08),
circle(1,11,0.10)
)}}} Graph of {{{-7x+6y=59}}} (red) and {{{-8x+6y=58}}} (green)