A line has a slope of `3/4` and intersects the points (-12, -39). At which point does this line intersect the x-axis?

Answer

To find the point where the line intersects the x-axis, we can follow these steps:

### Step 1: Write the equation of the line
We know the slope (m) of the line is \( \frac{3}{4} \) and it passes through the point (-12, -39). The equation of a line in slope-intercept form is given by:

\[ y = mx + c \]

### Step 2: Substitute the known values into the equation
We can substitute the slope and the coordinates of the point into the equation to find the y-intercept (c). We have:

\[ -39 = \frac{3}{4}(-12) + c \]

### Step 3: Solve for c
Now we will solve for c:

1. Calculate \( \frac{3}{4} \times -12 \):
\[ \frac{3}{4} \times -12 = -9 \]

2. Substitute this value back into the equation:
\[ -39 = -9 + c \]

3. Rearranging gives:
\[ c = -39 + 9 = -30 \]

### Step 4: Write the complete equation of the line
Now that we have the y-intercept, we can write the complete equation of the line:

\[ y = \frac{3}{4}x - 30 \]

### Step 5: Find the intersection with the x-axis
To find where the line intersects the x-axis, we set \( y = 0 \):

\[ 0 = \frac{3}{4}x - 30 \]

### Step 6: Solve for x
Now we solve for x:

1. Rearranging gives:
\[ \frac{3}{4}x = 30 \]

2. Multiply both sides by \( \frac{4}{3} \):
\[ x = 30 \times \frac{4}{3} = 40 \]

### Step 7: Write the coordinates of the intersection point
Thus, the point where the line intersects the x-axis is:

\[ (40, 0) \]

### Summary of the solution
The line intersects the x-axis at the point \( (40, 0) \).

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