Answer :
To graph the equation [tex]\( 9x - 6y = 18 \)[/tex] using intercepts, follow these steps:
### Step 1: Find the x-intercept
The x-intercept is where the graph crosses the x-axis, which means [tex]\( y = 0 \)[/tex].
1. Substitute [tex]\( y = 0 \)[/tex] into the equation:
[tex]\[ 9x - 6(0) = 18 \implies 9x = 18 \][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{18}{9} = 2 \][/tex]
So, the x-intercept is [tex]\((2, 0)\)[/tex].
### Step 2: Find the y-intercept
The y-intercept is where the graph crosses the y-axis, which means [tex]\( x = 0 \)[/tex].
1. Substitute [tex]\( x = 0 \)[/tex] into the equation:
[tex]\[ 9(0) - 6y = 18 \implies -6y = 18 \][/tex]
2. Solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{18}{-6} = -3 \][/tex]
So, the y-intercept is [tex]\((0, -3)\)[/tex].
### Step 3: Plot the intercepts
1. Plot the point [tex]\((2, 0)\)[/tex] on the x-axis.
2. Plot the point [tex]\((0, -3)\)[/tex] on the y-axis.
### Step 4: Draw the line
1. Draw a straight line through the two plotted points [tex]\((2, 0)\)[/tex] and [tex]\((0, -3)\)[/tex].
This line represents the equation [tex]\( 9x - 6y = 18 \)[/tex].
### Verifying with Slope-Intercept Form
We can also convert the equation to the slope-intercept form [tex]\( y = mx + b \)[/tex] to confirm our points and facilitate graphing:
1. Rewrite the original equation [tex]\( 9x - 6y = 18 \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ 9x - 6y = 18 \implies -6y = -9x + 18 \implies y = \frac{9x - 18}{6} \implies y = \frac{3}{2}x - 3 \][/tex]
This confirms the y-intercept [tex]\((0, -3)\)[/tex] and gives us the slope [tex]\(\frac{3}{2}\)[/tex], which can be used to verify the line through another method.
### Summary
- The x-intercept is [tex]\((2, 0)\)[/tex].
- The y-intercept is [tex]\((0, -3)\)[/tex].
- Plot these points on a graph.
- Draw a line through these points to graph the equation.
Remember to label the axes and provide a title for the graph to clearly illustrate the plotted line.
### Step 1: Find the x-intercept
The x-intercept is where the graph crosses the x-axis, which means [tex]\( y = 0 \)[/tex].
1. Substitute [tex]\( y = 0 \)[/tex] into the equation:
[tex]\[ 9x - 6(0) = 18 \implies 9x = 18 \][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{18}{9} = 2 \][/tex]
So, the x-intercept is [tex]\((2, 0)\)[/tex].
### Step 2: Find the y-intercept
The y-intercept is where the graph crosses the y-axis, which means [tex]\( x = 0 \)[/tex].
1. Substitute [tex]\( x = 0 \)[/tex] into the equation:
[tex]\[ 9(0) - 6y = 18 \implies -6y = 18 \][/tex]
2. Solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{18}{-6} = -3 \][/tex]
So, the y-intercept is [tex]\((0, -3)\)[/tex].
### Step 3: Plot the intercepts
1. Plot the point [tex]\((2, 0)\)[/tex] on the x-axis.
2. Plot the point [tex]\((0, -3)\)[/tex] on the y-axis.
### Step 4: Draw the line
1. Draw a straight line through the two plotted points [tex]\((2, 0)\)[/tex] and [tex]\((0, -3)\)[/tex].
This line represents the equation [tex]\( 9x - 6y = 18 \)[/tex].
### Verifying with Slope-Intercept Form
We can also convert the equation to the slope-intercept form [tex]\( y = mx + b \)[/tex] to confirm our points and facilitate graphing:
1. Rewrite the original equation [tex]\( 9x - 6y = 18 \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ 9x - 6y = 18 \implies -6y = -9x + 18 \implies y = \frac{9x - 18}{6} \implies y = \frac{3}{2}x - 3 \][/tex]
This confirms the y-intercept [tex]\((0, -3)\)[/tex] and gives us the slope [tex]\(\frac{3}{2}\)[/tex], which can be used to verify the line through another method.
### Summary
- The x-intercept is [tex]\((2, 0)\)[/tex].
- The y-intercept is [tex]\((0, -3)\)[/tex].
- Plot these points on a graph.
- Draw a line through these points to graph the equation.
Remember to label the axes and provide a title for the graph to clearly illustrate the plotted line.
