Question

y = -x - 1
y = 2x - 7
plot two lines by clicking the graph.
click a line to delete it.

Explanation:

To plot the lines \( y = -x - 1 \) and \( y = 2x - 7 \), we can use the slope - intercept form \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept.

For the line \( y=-x - 1 \)

• Step 1: Identify the y - intercept and slope

The equation is in the form \( y = mx + b \), where \( m=- 1\) (slope) and \( b=-1\) (y - intercept).
Start by plotting the y - intercept. The y - intercept is the point where \( x = 0 \), so the point is \( (0,-1) \).

• Step 2: Use the slope to find another point

The slope \( m=-1=\frac{-1}{1}\). From the point \( (0,-1) \), we move 1 unit down (because the numerator of the slope is - 1) and 1 unit to the right (because the denominator of the slope is 1). So we get the point \( (0 + 1,-1-1)=(1,-2) \). We can also move 1 unit up and 1 unit to the left: from \( (0,-1) \), moving 1 unit up (since - 1+1 = 0) and 1 unit to the left (since \( 0 - 1=-1\)) gives the point \( (-1,0) \).

For the line \( y = 2x-7 \)

• Step 1: Identify the y - intercept and slope

The equation is in the form \( y=mx + b \), where \( m = 2\) (slope) and \( b=-7\) (y - intercept).
Plot the y - intercept first. When \( x = 0 \), \( y=-7 \), so the point is \( (0,-7) \).

• Step 2: Use the slope to find another point

The slope \( m = 2=\frac{2}{1}\). From the point \( (0,-7) \), we move 2 units up (because the numerator of the slope is 2) and 1 unit to the right (because the denominator of the slope is 1). So we get the point \( (0 + 1,-7 + 2)=(1,-5) \). We can also move 2 units down and 1 unit to the left: from \( (0,-7) \), moving 2 units down (since \(-7-2=-9\)) and 1 unit to the left (since \( 0 - 1=-1\)) gives the point \( (-1,-9) \).

To plot the lines on the given graph:

• For \( y=-x - 1 \), plot the points \( (0,-1) \) and \( (1,-2) \) (or other points we found) and draw a line through them.
• For \( y = 2x-7 \), plot the points \( (0,-7) \) and \( (1,-5) \) (or other points we found) and draw a line through them.

If we want to find the intersection point of the two lines (to check our plotting), we can set the two equations equal to each other:
\(-x - 1=2x-7\)
Add \( x \) to both sides: \(-1=3x - 7\)
Add 7 to both sides: \( 6 = 3x\)
Divide both sides by 3: \( x = 2\)
Substitute \( x = 2\) into \( y=-x - 1 \): \( y=-2 - 1=-3\)
So the intersection point is \( (2,-3) \). We can check if this point lies on both lines:

• For \( y=-x - 1 \): When \( x = 2 \), \( y=-2 - 1=-3\) (correct).
• For \( y = 2x-7 \): When \( x = 2 \), \( y=2\times2-7 = 4 - 7=-3\) (correct).

To plot the lines:

1. For \( y=-x - 1 \):

• Click on the point \( (0,-1) \) (on the y - axis, 1 unit below the origin) and then click on the point \( (1,-2) \) (1 unit to the right of the origin and 2 units below the origin) to draw the line.

1. For \( y = 2x-7 \):

• Click on the point \( (0,-7) \) (7 units below the origin on the y - axis) and then click on the point \( (1,-5) \) (1 unit to the right of the origin and 5 units below the origin) to draw the line.

Answer:

To plot the lines \( y = -x - 1 \) and \( y = 2x - 7 \), we can use the slope - intercept form \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept.

For the line \( y=-x - 1 \)

• Step 1: Identify the y - intercept and slope

The equation is in the form \( y = mx + b \), where \( m=- 1\) (slope) and \( b=-1\) (y - intercept).
Start by plotting the y - intercept. The y - intercept is the point where \( x = 0 \), so the point is \( (0,-1) \).

• Step 2: Use the slope to find another point

The slope \( m=-1=\frac{-1}{1}\). From the point \( (0,-1) \), we move 1 unit down (because the numerator of the slope is - 1) and 1 unit to the right (because the denominator of the slope is 1). So we get the point \( (0 + 1,-1-1)=(1,-2) \). We can also move 1 unit up and 1 unit to the left: from \( (0,-1) \), moving 1 unit up (since - 1+1 = 0) and 1 unit to the left (since \( 0 - 1=-1\)) gives the point \( (-1,0) \).

For the line \( y = 2x-7 \)

• Step 1: Identify the y - intercept and slope

The equation is in the form \( y=mx + b \), where \( m = 2\) (slope) and \( b=-7\) (y - intercept).
Plot the y - intercept first. When \( x = 0 \), \( y=-7 \), so the point is \( (0,-7) \).

• Step 2: Use the slope to find another point

The slope \( m = 2=\frac{2}{1}\). From the point \( (0,-7) \), we move 2 units up (because the numerator of the slope is 2) and 1 unit to the right (because the denominator of the slope is 1). So we get the point \( (0 + 1,-7 + 2)=(1,-5) \). We can also move 2 units down and 1 unit to the left: from \( (0,-7) \), moving 2 units down (since \(-7-2=-9\)) and 1 unit to the left (since \( 0 - 1=-1\)) gives the point \( (-1,-9) \).

To plot the lines on the given graph:

• For \( y=-x - 1 \), plot the points \( (0,-1) \) and \( (1,-2) \) (or other points we found) and draw a line through them.
• For \( y = 2x-7 \), plot the points \( (0,-7) \) and \( (1,-5) \) (or other points we found) and draw a line through them.

If we want to find the intersection point of the two lines (to check our plotting), we can set the two equations equal to each other:
\(-x - 1=2x-7\)
Add \( x \) to both sides: \(-1=3x - 7\)
Add 7 to both sides: \( 6 = 3x\)
Divide both sides by 3: \( x = 2\)
Substitute \( x = 2\) into \( y=-x - 1 \): \( y=-2 - 1=-3\)
So the intersection point is \( (2,-3) \). We can check if this point lies on both lines:

• For \( y=-x - 1 \): When \( x = 2 \), \( y=-2 - 1=-3\) (correct).
• For \( y = 2x-7 \): When \( x = 2 \), \( y=2\times2-7 = 4 - 7=-3\) (correct).

To plot the lines:

1. For \( y=-x - 1 \):

• Click on the point \( (0,-1) \) (on the y - axis, 1 unit below the origin) and then click on the point \( (1,-2) \) (1 unit to the right of the origin and 2 units below the origin) to draw the line.

1. For \( y = 2x-7 \):

• Click on the point \( (0,-7) \) (7 units below the origin on the y - axis) and then click on the point \( (1,-5) \) (1 unit to the right of the origin and 5 units below the origin) to draw the line.