Question

9 lines l, m, n, and j are shown in the xy - coordinate plane. determine which line matches each equation (1) y=-1 + x (2) y=-4-\frac{1}{3}x (3) y = 2-3x (4) y=2-\frac{1}{2}x 0. use the digits 1 - 9 at most one time each to fill in both fractions so that the line has a slope of \frac{2}{3} and a y - intercept of 3 y = -x+ -

Explanation:

Step1: Recall slope - intercept form

The equation of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept.

Step2: Analyze equation (1) $y=-1 + x$

Here, $m = 1$ (positive slope) and $b=-1$. A line with a positive slope rises from left to right and crosses the y - axis at $y = - 1$.

Step3: Analyze equation (2) $y=-4-\frac{1}{3}x$

Here, $m=-\frac{1}{3}$ (negative slope) and $b = - 4$. A line with a negative slope falls from left to right and crosses the y - axis at $y=-4$.

Step4: Analyze equation (3) $y = 2-3x$

Here, $m=-3$ (negative slope) and $b = 2$. The line falls from left to right and crosses the y - axis at $y = 2$. Since the absolute value of the slope $|m| = 3$ is relatively large, the line is steeper among the negative - slope lines.

Step5: Analyze equation (4) $y=2-\frac{1}{2}x$

Here, $m=-\frac{1}{2}$ (negative slope) and $b = 2$. The line falls from left to right and crosses the y - axis at $y = 2$. Since $|-\frac{1}{2}|\lt| - 3|$, this line is less steep than the line $y = 2-3x$.

For the second part:
We know the equation of the line is $y=mx + b$, and we want $m=\frac{2}{3}$ and $b = 3$. So the equation is $y=\frac{2}{3}x+3$. We can write it as $y=\frac{4}{6}x + 3$ (using digits 1 - 9 without repetition, for example, we can use 4 for the numerator and 6 for the denominator of the slope fraction).

Answer:

For the first part, you need to match with the given graph (not provided in detail here for visual - matching). For the second part: $y=\frac{4}{6}x + 3$