Question

15 graph the given system of linear equations. ( 18x + 3y = -9 ) ( x - y = -4 ) choose the correct line to graph: line 1 is line ab. line 2 is line cd

Explanation:

Step1: Convert to slope - intercept form ($y = mx + b$) for the first equation

Given the equation $18x + 3y=-9$. We want to solve for $y$.
Subtract $18x$ from both sides: $3y=-18x - 9$.
Divide each term by 3: $y=\frac{-18x}{3}-\frac{9}{3}$, which simplifies to $y = - 6x-3$.

Step2: Convert to slope - intercept form for the second equation

Given the equation $x - y=-4$. We want to solve for $y$.
Subtract $x$ from both sides: $-y=-x - 4$.
Multiply both sides by - 1: $y=x + 4$.

Step3: Analyze the lines

For the line $y=-6x - 3$, the slope $m=-6$ (a steep negative slope) and the y - intercept $b = - 3$.
For the line $y=x + 4$, the slope $m = 1$ (a positive slope) and the y - intercept $b = 4$.

To graph these lines:

• For $y=-6x-3$: Start at the y - intercept $(0,-3)$. Then, using the slope (rise over run), since the slope is - 6 (or $\frac{-6}{1}$), from the point $(0,-3)$, go down 6 units and right 1 unit (or up 6 units and left 1 unit) to find other points.
• For $y=x + 4$: Start at the y - intercept $(0,4)$. Then, using the slope of 1 (or $\frac{1}{1}$), from the point $(0,4)$, go up 1 unit and right 1 unit (or down 1 unit and left 1 unit) to find other points.

If we assume Line AB and Line CD correspond to these two lines:

• The line with equation $y=-6x - 3$ will have a steeper negative slope.
• The line with equation $y=x + 4$ will have a positive slope.

To determine which is Line AB and which is Line CD, we can also find two points on each line:

For $18x + 3y=-9$ (or $y=-6x - 3$):
When $x = 0$, $y=-3$. When $x=-0.5$, $y=-6\times(-0.5)-3=3 - 3=0$. So two points are $(0,-3)$ and $(-0.5,0)$.

For $x - y=-4$ (or $y=x + 4$):
When $x = 0$, $y = 4$. When $x=-4$, $y=-4 + 4=0$. So two points are $(0,4)$ and $(-4,0)$.

To graph the system, we plot these two lines. The line with the equation $y=-6x - 3$ (from $18x + 3y=-9$) and the line with the equation $y=x + 4$ (from $x - y=-4$) are the two lines of the system.

If we need to match to Line AB and Line CD:

• Line with $y=x + 4$ (slope 1, y - intercept 4) will have a positive slope and pass through $(0,4)$ and $(-4,0)$.
• Line with $y=-6x - 3$ (slope - 6, y - intercept - 3) will have a steep negative slope and pass through $(0,-3)$ and $(-0.5,0)$.

(Note: Since the problem mentions "Choose the correct line to graph: Line 1 is Line AB. Line 2 is Line CD", and we have two lines from the system, we can conclude that Line AB and Line CD are the two lines represented by $18x + 3y=-9$ and $x - y=-4$ respectively (or vice - versa depending on the graphing tool's setup). But the main task is to graph the two lines by converting them to slope - intercept form and using the slope and y - intercept to plot the points.)

Answer:

To graph the system

$$\begin{cases}18x + 3y=-9\\x - y=-4\end{cases}$$

, we convert to slope - intercept form: $y=-6x - 3$ (from $18x + 3y=-9$) and $y=x + 4$ (from $x - y=-4$). Plot the line $y=-6x - 3$ (with slope - 6, y - intercept - 3) and the line $y=x + 4$ (with slope 1, y - intercept 4) on the coordinate plane. If Line AB and Line CD are these two lines, Line AB can be one of them and Line CD the other, depending on the slope and intercept characteristics.