Sure! Let's start with the given equation and transform it step-by-step into the slope-intercept form:
### Step-by-Step Solution:
1. Start with the Given Equation:
[tex]\[ 5x + 10y = -30 \][/tex]
2. Isolate the Term with [tex]\( y \)[/tex]:
We want to get [tex]\( y \)[/tex] by itself on one side of the equation. First, subtract [tex]\( 5x \)[/tex] from both sides:
[tex]\[ 10y = -5x - 30 \][/tex]
3. Solve for [tex]\( y \)[/tex]:
Now, to isolate [tex]\( y \)[/tex], divide every term in the equation by 10:
[tex]\[ y = \frac{-5x}{10} - \frac{30}{10} \][/tex]
4. Simplify the Equation:
Simplify the fractions on the right-hand side of the equation:
[tex]\[ y = \frac{-5}{10}x - \frac{30}{10} \][/tex]
[tex]\[ y = -\frac{1}{2}x - 3 \][/tex]
The equation in slope-intercept form is:
[tex]\[ y = -\frac{1}{2}x - 3 \][/tex]
### 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 y-intercept.
From our equation:
- The slope ([tex]\( m \)[/tex]) = -[tex]\(\frac{1}{2}\)[/tex]
- The y-intercept ([tex]\( b \)[/tex]) = -3
### Graphing the Equation:
To graph the equation [tex]\( y = -\frac{1}{2}x - 3 \)[/tex], follow these steps:
1. Plot the Y-Intercept:
Start by plotting the y-intercept [tex]\( b \)[/tex]. For this equation, the y-intercept is -3, so place a point at (0, -3) on the coordinate plane.
2. Use the Slope:
The slope of -[tex]\(\frac{1}{2}\)[/tex] means that for every 1 unit you move to the right on the x-axis, you move down [tex]\( \frac{1}{2} \)[/tex] unit on the y-axis (since it's a negative slope).
- Starting from the y-intercept point (0, -3), move 1 unit to the right (to x = 1).
- From there, move down [tex]\( \frac{1}{2} \)[/tex] unit to reach the point (1, -3.5).
3. Draw the Line:
Draw a line through the points you have plotted. This line represents the graph of the equation [tex]\( y = -\frac{1}{2}x - 3 \)[/tex].
Following these steps will allow you to successfully graph the equation [tex]\( 5x + 10y = -30 \)[/tex] in the slope-intercept form.
