Question 203093
# 1



Start with the given system of equations:



{{{system(x+y=10,y=x+8)}}}



{{{y=x+8}}} Start with the second equation.



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



{{{x+y=10}}} Move onto the first equation.



{{{x+x+8=10}}} Plug in {{{y=x+8}}}.



{{{2x+8=10}}} Combine like terms on the left side.



{{{2x=10-8}}} Subtract {{{8}}} from both sides.



{{{2x=2}}} Combine like terms on the right side.



{{{x=(2)/(2)}}} Divide both sides by {{{2}}} to isolate {{{x}}}.



{{{x=1}}} Reduce.



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



Since we know that {{{x=1}}}, we can use this to find {{{y}}}.



{{{y=x+8}}} Go back to the second equation.



{{{y=1+8}}} Plug in {{{x=1}}}.



{{{y=9}}} Add



So the solutions are {{{x=1}}} and {{{y=9}}}.



This means that the system is consistent and independent.



Notice when we graph the equations, we see that they intersect at *[Tex \LARGE \left(1,9\right)]. So this visually verifies our answer.



{{{drawing(500,500,-9,11,-1,19,
grid(1),
graph(500,500,-9,11,-1,19,10-x,x+8),
circle(1,9,0.05),
circle(1,9,0.08),
circle(1,9,0.10)
)}}} Graph of {{{x+y=10}}} (red) and {{{y=x+8}}} (green) 



<hr>


# 2




Start with the given system of equations:



{{{system(3x+y=5,4x-7y=-10)}}}



{{{3x+y=5}}} Start with the first equation.



{{{y=5-3x}}} Subtract {{{3x}}} from both sides.



{{{y=-3x+5}}} Rearrange the terms and simplify.



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



{{{4x-7y=-10}}} Move onto the second equation.



{{{4x-7(-3x+5)=-10}}} Now plug in {{{y=-3x+5}}}.



{{{4x+21x-35=-10}}} Distribute.



{{{25x-35=-10}}} Combine like terms on the left side.



{{{25x=-10+35}}} Add {{{35}}} to both sides.



{{{25x=25}}} Combine like terms on the right side.



{{{x=(25)/(25)}}} Divide both sides by {{{25}}} to isolate {{{x}}}.



{{{x=1}}} Reduce.



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



Since we know that {{{x=1}}}, we can use this to find {{{y}}}.



{{{3x+y=5}}} Go back to the first equation.



{{{3(1)+y=5}}} Plug in {{{x=1}}}.



{{{3+y=5}}} Multiply.



{{{y=5-3}}} Subtract {{{3}}} from both sides.



{{{y=2}}} Combine like terms on the right side.



So the solutions are {{{x=1}}} and {{{y=2}}}.



This means that the system is consistent and independent.



Notice when we graph the equations, we see that they intersect at *[Tex \LARGE \left(1,2\right)]. So this visually verifies our answer.



{{{drawing(500,500,-9,11,-8,12,
grid(1),
graph(500,500,-9,11,-8,12,5-3x,(-10-4x)/(-7)),
circle(1,2,0.05),
circle(1,2,0.08),
circle(1,2,0.10)
)}}} Graph of {{{3x+y=5}}} (red) and {{{4x-7y=-10}}} (green) 



<hr>


# 3




Start with the given system of equations:



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



{{{-2x+y=-5}}} Start with the first equation.



{{{y=-5+2x}}} Add {{{2x}}} to both sides.



{{{y=2x-5}}} Rearrange the terms and simplify.



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



{{{-x+3y=5}}} Move onto the second equation.



{{{-x+3(2x-5)=5}}} Now plug in {{{y=2x-5}}}.



{{{-x+6x-15=5}}} Distribute.



{{{5x-15=5}}} Combine like terms on the left side.



{{{5x=5+15}}} Add {{{15}}} to both sides.



{{{5x=20}}} Combine like terms on the right side.



{{{x=(20)/(5)}}} Divide both sides by {{{5}}} to isolate {{{x}}}.



{{{x=4}}} Reduce.



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



Since we know that {{{x=4}}}, we can use this to find {{{y}}}.



{{{-2x+y=-5}}} Go back to the first equation.



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



{{{-8+y=-5}}} Multiply.



{{{y=-5+8}}} Add {{{8}}} to both sides.



{{{y=3}}} Combine like terms on the right side.



So the solutions are {{{x=4}}} and {{{y=3}}}.



This means that the system is consistent and independent.



Notice when we graph the equations, we see that they intersect at *[Tex \LARGE \left(4,3\right)]. So this visually verifies our answer.



{{{drawing(500,500,-6,14,-7,13,
grid(1),
graph(500,500,-6,14,-7,13,-5+2x,(5+x)/(3)),
circle(4,3,0.05),
circle(4,3,0.08),
circle(4,3,0.10)
)}}} Graph of {{{-2x+y=-5}}} (red) and {{{-x+3y=5}}} (green)