I have this problem: y=1x and y=.5x+4
I got (2,4) and was wondering if you guys can show me the steps for both elimination and substitution. Also how would I graph this? just put one dot on the graph?
How did you get $\displaystyle (2,4)?$ If you try checking your work by substituting, you get
$\displaystyle x=2\Rightarrow y=2$
for the first, and
$\displaystyle x=2\Rightarrow y=5$
for the second. Your point is not on either of the curves, so it certainly could not be in the solution set.
Certainly. We have
$\displaystyle \left\{\begin{array}{rcl}
y&=&x\\
y&=&\frac12x+4
\end{array}\right.$
Substitution:
Substituting $\displaystyle x=y$ into the second equation produces
$\displaystyle y=\frac12y+4,$
and solving for $\displaystyle y$ gives
$\displaystyle \frac12y=4\Rightarrow y=8.$
Back-substituting, we get $\displaystyle x=8.$
Elimination:
Let's first rearrange the equations a little,
$\displaystyle \left\{\begin{array}{rcl}
y-x&=&0\\
2y-x&=&8
\end{array}\right..$
Subtract the first equation from the second,
$\displaystyle \left\{\begin{array}{rcl}
y-x&=&0\\
y&=&8
\end{array}\right.$
and subtract the second equation from the first:
$\displaystyle \left\{\begin{array}{rcl}
-x&=&-8\\
y&=&8
\end{array}\right.\Rightarrow\left\{\begin{array}{ rcl}
x&=&8\\
y&=&8
\end{array}\right.$