Answer:The tricky thing is that there are two variables, xxxx and yyyy. If only we could get rid of one of the variables...
Here's an idea! Equation 1111 tells us that 2x\goldD{2x}2xstart color #e07d10, 2, x, end color #e07d10 and y\goldD yystart color #e07d10, y, end color #e07d10 are equal. So let's plug in 2x\goldD{2x}2xstart color #e07d10, 2, x, end color #e07d10 for y\goldD yystart color #e07d10, y, end color #e07d10 in Equation 2222 to get rid of the yyyy variable in that equation:
x+y=24Equation 2x+2x=24Substitute 2x for y\begin{aligned} x + \goldD y &= 24 &\gray{\text{Equation 2}} \\\\ x + \goldD{2x} &= 24 &\gray{\text{Substitute 2x for y}}\end{aligned}x+yx+2x=24=24Equation 2Substitute 2x for y
Brilliant! Now we have an equation with just the xxxx variable that we know how to solve:
x+2x3x 3x3x=24=24=243=8Divide each side by 3
Nice! So we know that xxxx equals 8888. But remember that we are looking for an ordered pair. We need a yyyy value as well. Let's use the first equation to find yyyy when xxxx equals 8888:
y=2xEquation 1y=2(8)Substitute 8 for xy=16\begin{aligned} y &= 2\blueD x &\gray{\text{Equation 1}} \\\\ y &= 2(\blueD8) &\gray{\text{Substitute 8 for x}}\\\\ \greenD y &\greenD= \greenD{16}\end{aligned}yyy=2x=2(8)=16Equation 1Substitute 8 for x
Sweet! So the solution to the system of equations is (8,16)(\blueD8, \greenD{16})(8,16)left parenthesis, start color #11accd, 8, end color #11accd, comma, start color #1fab54, 16, end color #1fab54, right parenthesis. It's always a good idea to check the solution back in the original equations just to be sure.
Let's check the first equation:
y=2x16=?2(8)Plug in x = 8 and y = 1616=16Yes!\begin{aligned} y &= 2x \\\\ \greenD{16} &\stackrel?= 2(\blueD{8}) &\gray{\text{Plug in x = 8 and y = 16}}\\\\ 16 &= 16 &\gray{\text{Yes!}}\end{aligned}y1616=2x=?2(8)=16Plug in x = 8 and y = 16Yes!
Let's check the second equation:
x+y=248+16=?24Plug in x = 8 and y = 1624=24Yes!\begin{aligned} x +y &= 24 \\\\ \blueD{8} + \greenD{16} &\stackrel?= 24 &\gray{\text{Plug in x = 8 and y = 16}}\\\\ 24 &= 24 &\gray{\text{Yes!}}\end{aligned}x+y8+1624=24=?24=24Plug in x = 8 and y = 16Yes!
Great! (8,16)(\blueD8, \greenD{16})(8,16)left parenthesis, start color #11accd, 8, end color #11accd, comma, start color #1fab54, 16, end color #1fab54, right parenthesis is indeed a solution. We must not have made any mistakes.
Your turn to solve a system of equations using substitution.
Use substitution to solve the following system of equations.
4x+y=284x + y = 284x+y=284, x, plus, y, equals, 28
y=3xy = 3xy=3x
