To determine the intercepts of the line given by the equation [tex]\(4x - 1 = 3y + 5\)[/tex], we need to find both the [tex]\(x\)[/tex]-intercept and the [tex]\(y\)[/tex]-intercept. Let's go through the steps to find each intercept.
### Finding the [tex]\(x\)[/tex]-Intercept
The [tex]\(x\)[/tex]-intercept is the point where the line crosses the [tex]\(x\)[/tex]-axis. This occurs when [tex]\(y = 0\)[/tex]. To find the [tex]\(x\)[/tex]-intercept, we substitute [tex]\(y = 0\)[/tex] into the equation and solve for [tex]\(x\)[/tex]:
1. Start with the original equation:
[tex]\[ 4x - 1 = 3y + 5 \][/tex]
2. Substitute [tex]\(y = 0\)[/tex]:
[tex]\[ 4x - 1 = 3(0) + 5 \][/tex]
3. Simplify the equation:
[tex]\[ 4x - 1 = 5 \][/tex]
4. Add 1 to both sides:
[tex]\[ 4x = 6 \][/tex]
5. Divide by 4:
[tex]\[ x = \frac{6}{4} = 1.5 \][/tex]
Thus, the [tex]\(x\)[/tex]-intercept is [tex]\( x = 1.5 \)[/tex].
### Finding the [tex]\(y\)[/tex]-Intercept
The [tex]\(y\)[/tex]-intercept is the point where the line crosses the [tex]\(y\)[/tex]-axis. This occurs when [tex]\(x = 0\)[/tex]. To find the [tex]\(y\)[/tex]-intercept, we substitute [tex]\(x = 0\)[/tex] into the equation and solve for [tex]\(y\)[/tex]:
1. Start with the original equation:
[tex]\[ 4x - 1 = 3y + 5 \][/tex]
2. Substitute [tex]\(x = 0\)[/tex]:
[tex]\[ 4(0) - 1 = 3y + 5 \][/tex]
3. Simplify the equation:
[tex]\[ -1 = 3y + 5 \][/tex]
4. Subtract 5 from both sides:
[tex]\[ -6 = 3y \][/tex]
5. Divide by 3:
[tex]\[ y = \frac{-6}{3} = -2 \][/tex]
Thus, the [tex]\(y\)[/tex]-intercept is [tex]\( y = -2 \)[/tex].
### Summary
- [tex]\(x\)[/tex]-intercept: [tex]\( 1.5 \)[/tex]
- [tex]\(y\)[/tex]-intercept: [tex]\( -2 \)[/tex]
