Question

y = x - 2\
y = -3x - 6\
plot two lines by clicking the graph.\
click a line to delete it.

Explanation:

To plot the lines \( y = x - 2 \) and \( y = -3x - 6 \), we can use the slope - intercept form \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept.

For the line \( y=x - 2 \)

Step 1: Identify slope and y - intercept

The equation is in the form \( y=mx + b \), where the slope \( m = 1\) and the y - intercept \( b=- 2\).

• To find the y - intercept point, we set \( x = 0\). Substituting \( x = 0\) into \( y=x - 2\), we get \( y=0 - 2=-2\). So the point is \( (0,-2) \).
• To find another point, we can use the slope. The slope \( m = 1=\frac{1}{1}\), which means for a change in \( x\) of \( 1\) unit (we can move \( x\) by \( 1\) to the right), the change in \( y\) is \( 1\) unit (we move \( y\) up by \( 1\)). Starting from \( (0,-2) \), if we move \( x = 0+1 = 1\) and \( y=-2 + 1=-1\), we get the point \( (1,-1) \).

For the line \( y=-3x - 6 \)

Step 1: Identify slope and y - intercept

The equation is in the form \( y = mx + b\), where the slope \( m=-3\) and the y - intercept \( b = - 6\).

• To find the y - intercept point, we set \( x = 0\). Substituting \( x = 0\) into \( y=-3x - 6\), we get \( y=0-6=-6\). So the point is \( (0,-6) \).
• To find another point, we use the slope. The slope \( m=-3=\frac{-3}{1}\), which means for a change in \( x\) of \( 1\) unit (move \( x\) to the right by \( 1\)), the change in \( y\) is \( - 3\) units (move \( y\) down by \( 3\)). Starting from \( (0,-6) \), if we move \( x=0 + 1=1\) and \( y=-6-3=-9\), we get the point \( (1,-9) \).

To plot the lines:

• For \( y=x - 2\), plot the points \( (0,-2) \) and \( (1,-1) \) (and other points if needed) and draw a straight line through them.
• For \( y=-3x - 6\), plot the points \( (0,-6) \) and \( (1,-9) \) (and other points if needed) and draw a straight line through them.

(If we want to find the intersection point of the two lines, we can set the two equations equal to each other:
\(x - 2=-3x - 6\)

Step 1: Solve for \( x\)

Add \( 3x\) to both sides of the equation: \(x+3x - 2=-3x+3x - 6\), which simplifies to \(4x-2=-6\).
Then add \( 2\) to both sides: \(4x-2 + 2=-6 + 2\), so \(4x=-4\).
Divide both sides by \( 4\): \(x=\frac{-4}{4}=-1\).

Step 2: Solve for \( y\)

Substitute \( x = - 1\) into \( y=x - 2\) (we could also use the other equation). Then \( y=-1-2=-3\). So the intersection point of the two lines is \( (-1,-3) \))

If the task is to plot the lines, the key points for plotting are:

• Line \( y = x-2\): passes through \( (0,-2)\) and \( (1,-1)\)
• Line \( y=-3x - 6\): passes through \( (0,-6)\) and \( (1,-9)\)

If the task is to find the intersection:

Answer:

The intersection point of the lines \( y=x - 2\) and \( y=-3x - 6\) is \( (-1,-3) \)