SAT Mathematics Practice Test 2

Since the slope of the first line is 2, an equation of this line can be written in the form $\mathit{y}=2\mathit{x}+\mathit{c}$,where c is the y-intercept of the line. Since the line contains the point $\left(1,8\right)$,one can substitute 1 for x and 8 for y in $\mathit{y}=2\mathit{x}+\mathit{c}$, which gives $8=2\left(1\right)+\mathit{c}$, or $\mathit{c}=6$. Thus, an equation of the first line is $\mathit{y}=2\mathit{x}+6$.The slope of the second line is equal to $\frac{1-2}{2-1}$ or $-1$.Thus, an equation of the second line can be written in the form $\mathit{y}=-\mathit{x}+\mathit{d}$,where d is the y-intercept of the line. Substituting 2 for x and 1 for y gives $1=-2+\mathit{d}$,or$\mathit{d}=3$.Thus, an equation of the second line is $\mathit{y}=-\mathit{x}+3$.
Since a is the x-coordinate and b is the y-coordinate of the intersection point of the two lines, one can substitute a for x and b for y in the two equations, giving the system $\mathit{b}=2\mathit{a}+6$ and $\mathit{b}=\mathit{–}\mathit{a}+3$. Thus, a can be found by solving the equation $2\mathit{a}+6=-\mathit{a}+3$,which gives $\mathit{a}=-1.$ Finally, substituting $-1$ for a into the equation $\mathit{b}=\mathit{–}\mathit{a}+3$ gives $\mathit{b}=-\left(-1\right)+3$, or $\mathit{b}=4$.Therefore, the value of $\mathit{a}+\mathit{b}$ is $3$.
Alternatively, since the second line passes through the points $\left(1,2\right)$ and $\left(2,1\right)$, an equation for the second line is $\mathit{x}+\mathit{y}=3$. Thus, the intersection point of the first line and the second line, $\left(\mathit{a},\mathit{b}\right)$ lies on the line with equation $\mathit{x}+\mathit{y}=3$. It follows that $\mathit{a}+\mathit{b}=3$
Choices A and C are incorrect and may result from finding the value of only $\mathit{a}$ or $\mathit{b}$,but not calculating the value of $\mathit{a}+\mathit{b}$. Choice D is incorrect and may result from a computation error in finding equations of the two lines or in solving the resulting system of equations.