Certainly! Let's break this down step-by-step.
### Step 1: Identify the given equation
You are given the linear equation:
[tex]\[
-6x + 3y = 3
\][/tex]
### Step 2: Convert to slope-intercept form
The slope-intercept form of a linear equation is:
[tex]\[
y = mx + b
\][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the [tex]\( y \)[/tex]-intercept.
To convert the given equation to this form, we need to solve for [tex]\( y \)[/tex].
### Step 3: Isolate [tex]\( y \)[/tex]
[tex]\[
-6x + 3y = 3
\][/tex]
First, move the [tex]\( -6x \)[/tex] term to the right side of the equation:
[tex]\[
3y = 6x + 3
\][/tex]
Next, divide every term by 3 to solve for [tex]\( y \)[/tex]:
[tex]\[
y = 2x + 1
\][/tex]
### Step 4: Identify the slope and [tex]\( y \)[/tex]-intercept
Now that we have the equation in the form [tex]\( y = 2x + 1 \)[/tex], we can identify the slope and the [tex]\( y \)[/tex]-intercept.
The slope ([tex]\( m \)[/tex]) is the coefficient of [tex]\( x \)[/tex], which is:
[tex]\[
m = 2
\][/tex]
The [tex]\( y \)[/tex]-intercept ([tex]\( b \)[/tex]) is the constant term, which is:
[tex]\[
b = 1
\][/tex]
### Step 5: Graph the line
To graph the line, follow these steps:
1. Start with the [tex]\( y \)[/tex]-intercept ([tex]\( b \)[/tex]). Plot a point on the [tex]\( y \)[/tex]-axis at [tex]\( (0, 1) \)[/tex].
2. Use the slope ([tex]\( m = 2 \)[/tex]) to determine the next point. Slope is defined as the rise over the run. So, from the [tex]\( y \)[/tex]-intercept [tex]\( (0, 1) \)[/tex]:
- Go up 2 units (rise)
- Go right 1 unit (run)
This gives you another point on the line at [tex]\( (1, 3) \)[/tex].
3. Draw a straight line through these two points.
### Final Line Equation
The final equation, as converted to slope-intercept form, is:
[tex]\[
y = 2x + 1
\][/tex]
This line has a slope of [tex]\( 2 \)[/tex] and a [tex]\( y \)[/tex]-intercept of [tex]\( 1 \)[/tex].
