Question 154342


Start with the given system of equations:

{{{system(x+2y=5,3x+4y=1)}}}



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



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



So we have the new system of equations:

{{{system(-2x-4y=-10,3x+4y=1)}}}



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-4y)+(3x+4y)=(-10)+(1)}}}



{{{(-2x+3x)+(-4y+4y)=-10+1}}} Group like terms.



{{{x+0y=-9}}} Combine like terms. Notice how the y terms cancel out.



{{{x=-9}}} Simplify.



------------------------------------------------------------------



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



{{{-2(-9)-4y=-10}}} Plug in {{{x=-9}}}.



{{{18-4y=-10}}} Multiply.



{{{-4y=-10-18}}} Subtract {{{18}}} from both sides.



{{{-4y=-28}}} Combine like terms on the right side.



{{{y=(-28)/(-4)}}} Divide both sides by {{{-4}}} to isolate {{{y}}}.



{{{y=7}}} Reduce.



So our answer is {{{x=-9}}} and {{{y=7}}}.



Which form the ordered pair *[Tex \LARGE \left(-9,7\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(-9,7\right)]. So this visually verifies our answer.



{{{drawing(500,500,-19,1,-3,17,
grid(1),
graph(500,500,-19,1,-3,17,(5-x)/(2),(1-3x)/(4)),
circle(-9,7,0.05),
circle(-9,7,0.08),
circle(-9,7,0.10)
)}}} Graph of {{{x+2y=5}}} (red) and {{{3x+4y=1}}} (green)