Question

part 2 comprehension check
points: ____/

1. consider this system of equations.

$y = 4x$
$3x + 2y = 33$
solve the system of equations using substitution. (3 points)
step 1 solve
$3x + 2y = 33$
$3x + 2(\underline{quadquad}) = 33$
$3x + \underline{quadquad} = 33$
$11x = 33$
$x = \underline{quadquad}$
step 2 solve
$y = 4x$
$y = 4(\underline{quadquad})$
$y = \underline{quadquad}$
solution: $(\underline{quadquad}, \underline{quadquad})$

1. evan graphs the system of equations below.

$y = \frac{1}{3}x - 5$
$y = -\frac{1}{2}x + 4$
he says there is no solution because the lines do not intersect on his graph.
is evan correct? (3 points)
choose one option for each blank to answer the question.
evan \boxed{\quad} correct because two lines in a system with no solution have slopes that must be \boxed{\quad} and y-intercepts that \boxed{\quad}
\

$$\begin{tabular}{|c|} \\hline a is \\\\ b is not \\\\ \\hline \\end{tabular}$$

\

$$\begin{tabular}{|c|} \\hline a different \\\\ b the same \\\\ \\hline \\end{tabular}$$

\

$$\begin{tabular}{|c|} \\hline a must be different \\\\ b must be the same \\\\ c can be the same or different \\\\ \\hline \\end{tabular}$$

(graph: a coordinate plane with two lines, one with negative slope and one with positive slope, not intersecting in the visible grid area.)

Explanation:

Step1: Substitute $y=4x$ into equation

$3x + 2(4x) = 33$

Step2: Simplify left-hand side

$3x + 8x = 33$

Step3: Solve for $x$

$11x = 33 \implies x = \frac{33}{11} = 3$

Step4: Substitute $x=3$ into $y=4x$

$y = 4(3) = 12$

Step5: Evaluate Evan's claim

First, check slopes: $m_1=\frac{1}{3}$, $m_2=-\frac{1}{2}$ (different, so lines intersect, meaning Evan is wrong). Systems with no solution have equal slopes, different y-intercepts.

Answer:

1. Step1 blank: $4x$; Step1 blank: $8x$; $x=3$; $y=12$; Solution: $(3, 12)$
2. Evan $\boldsymbol{B. is not}$ correct because two lines in a system with no solution have slopes that must be $\boldsymbol{B. the same}$ and y-intercepts that $\boldsymbol{A. must be different}$