Let's determine the intercepts of the line given by the equation [tex]\( y = -3x + 12 \)[/tex].
### Finding the y-intercept:
The y-intercept occurs when [tex]\( x = 0 \)[/tex]. At this point, the value of [tex]\( y \)[/tex] can be calculated by substituting [tex]\( x = 0 \)[/tex] into the equation [tex]\( y = -3x + 12 \)[/tex].
1. Substitute [tex]\( x = 0 \)[/tex] into the equation:
[tex]\[
y = -3(0) + 12
\][/tex]
2. Simplify the equation:
[tex]\[
y = 12
\][/tex]
Therefore, the y-intercept of the line is [tex]\( (0, 12) \)[/tex].
### Finding the x-intercept:
The x-intercept occurs when [tex]\( y = 0 \)[/tex]. At this point, the value of [tex]\( x \)[/tex] can be calculated by setting [tex]\( y = 0 \)[/tex] in the equation [tex]\( y = -3x + 12 \)[/tex] and solving for [tex]\( x \)[/tex].
1. Set [tex]\( y = 0 \)[/tex] in the equation:
[tex]\[
0 = -3x + 12
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[
-3x + 12 = 0
\][/tex]
3. Subtract 12 from both sides:
[tex]\[
-3x = -12
\][/tex]
4. Divide both sides by -3:
[tex]\[
x = 4
\][/tex]
Therefore, the x-intercept of the line is [tex]\( (4.0, 0) \)[/tex].
So the intercepts of the line are:
- y-intercept: [tex]\( (0, 12) \)[/tex]
- x-intercept: [tex]\( (4.0, 0) \)[/tex]
Thus filling in the blanks, we get:
[tex]\[
\begin{array}{l}
y=-3 x+12 \\
y \text {-intercept: } (0, 12) \\
x \text {-intercept: } (4.0, 0)
\end{array}
\][/tex]
