Question

Write the equation for a line parallel to the line 3x−2y=10 and goes through point (-4, 6).

Write in slope-intercept form

Answer 1

Could you simplify it?

Answer 2

To write the equation of a line parallel to the line \(3x - 2y = 10\) that passes through the point \((-4, 6)\), we first need to determine the slope of the original line.

### Step 1: Find the slope of the given line

We can rearrange the equation \(3x - 2y = 10\) into slope-intercept form \(y = mx + b\), where \(m\) represents the slope.

Starting with the given equation:
\[
3x - 2y = 10
\]
We isolate \(y\):
\[
-2y = -3x + 10
\]
Dividing every term by \(-2\) gives:
\[
y = \frac{3}{2}x - 5
\]
From this, we see that the slope \(m\) of the line is \(\frac{3}{2}\).

### Step 2: Use the slope for the parallel line

Since parallel lines have the same slope, the slope of our new line will also be \(\frac{3}{2}\).

### Step 3: Use point-slope form to find the equation

We will use the point-slope form of the equation of a line, which is:
\[
y - y_1 = m(x - x_1)
\]
where \((x_1, y_1)\) is the point the line passes through, and \(m\) is the slope.

Substituting in the slope \(\frac{3}{2}\) and the point \((-4, 6)\):
\[
y - 6 = \frac{3}{2}(x + 4)
\]

### Step 4: Simplify to slope-intercept form

Now, we will simplify this equation. Expanding the right side:
\[
y - 6 = \frac{3}{2}x + \frac{3}{2} \times 4
\]
Calculating \(\frac{3}{2} \times 4\):
\[
\frac{3}{2} \times 4 = 6
\]
So we have:
\[
y - 6 = \frac{3}{2}x + 6
\]

Next, we add 6 to both sides:
\[
y = \frac{3}{2}x + 12
\]

### Final Equation

The equation of the line that is parallel to the line \(3x - 2y = 10\) and passes through the point \((-4, 6)\) is:
\[
\boxed{y = \frac{3}{2}x + 12}
\]

Answer 3

The equation \(y = \frac{3}{2}x + 12\) is already in slope-intercept form, which is the standard way to present the equation of a line.

However, if you would like to express it in a different but equivalent form (for example, removing the fraction), you can multiply the entire equation by 2 to eliminate the fraction:

Starting from:
\[
y = \frac{3}{2}x + 12
\]

Multiply every term by 2:
\[
2y = 3x + 24
\]

Now, rearranging this to standard form \(Ax + By = C\) gives:
\[
-3x + 2y = 24
\]

If we want a positive \(A\), we can multiply the entire equation by -1:
\[
3x - 2y = -24
\]

So, the equation can also be expressed in standard form as:
\[
\boxed{3x - 2y = -24}
\]

But note that depending on the context, the simplified version of the original slope-intercept form may be what you're looking for, so \(y = \frac{3}{2}x + 12\) is also perfectly valid!