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Quiz 11
1.) Find an equation for the line with slope $3$ and $y$-intercept $(0,5)$.
Solution: Using the formula $y=mx+b$ where $m$ denotes the slope and $b$ denotes the $y$-coordinate of the $y$-intercept, we have $m=3$ and $b=5$ and so the equation of the line is $$y=3x+5.$$i

2.) Find an equation of the line with slope $2$ passing through the point $(3,7)$.
Solution: Recall that the point-slope formula for the equation of a line with slope $m$ passing through the point $(x_1,y_1)$ is $$y-y_1=m(x-x_1).$$ For this particular line, $m=2$ and $(x_1,y_1)=(3,7)$. Therefore the equation of the line is $$y-7=2(x-3).$$

3.) Find an equation of the line passing through the points $(1,2)$ and $(3,4)$.
Solution: To find the equation of this line, we will want to use the point-slope form. To do that we must first find the slope of this line: $$\mathrm{slope}=m=\dfrac{4-2}{3-1}=\dfrac{2}{2}=1.$$ Now we will use point-slope form with $m=1$ and $(x_1,y_1)=(1,2)$ (note: we could have chose the other point too) and we get $$y-2=1(x-1),$$ or more simply $$y-2=x-1,$$ or equivalently $$y=x+1.$$

4.) Find an equation of the line passing through the points $(-1,3)$ and $(5,-9)$.
Solution: Find the slope: $$\mathrm{slope}=\dfrac{-9-3}{5-(-1)} = \dfrac{-12}{6} = -2.$$ Using point-slope form with $m=-2$ and $(x_1,y_1)=(-1,3)$, we get $$y-3=-2(x-(-1)),$$ or equivalently $$y-3=-2x-2,$$ or equivalently $$y=-2x+1.$$

5.) Simplify the expression $(-5)^2$.
Solution: $(-5)^2=(-5)(-5)=25$