Question

Find the equation in standard form of the line through -3, five and parallel to 2X minus 4Y equals three

Answer 1

Wrong

Answer 2

To find the equation of a line in standard form that is parallel to the given line \(2x - 4y = 3\) and passes through the point (-3, 5), we follow these steps:

1. **Find the slope of the line from the given equation.**
We will rewrite \(2x - 4y = 3\) in slope-intercept form (y = mx + b), where m is the slope.

Starting with the original equation:
\[
2x - 4y = 3
\]
Rearranging gives:
\[
-4y = -2x + 3
\]
Dividing by -4:
\[
y = \frac{1}{2}x - \frac{3}{4}
\]
The slope \(m\) of the line is \(\frac{1}{2}\).

2. **Use the slope point form to write the equation of the line.**
Since parallel lines have the same slope, the slope of our new line will also be \(\frac{1}{2}\). We can use the point-slope form of a line, which is:
\[
y - y_1 = m(x - x_1)
\]
Here, \((x_1, y_1) = (-3, 5)\) and \(m = \frac{1}{2}\):
\[
y - 5 = \frac{1}{2}(x + 3)
\]

Expanding this:
\[
y - 5 = \frac{1}{2}x + \frac{3}{2}
\]
Adding 5 to both sides yields:
\[
y = \frac{1}{2}x + \frac{3}{2} + 5
\]
Converting 5 to halves gives us:
\[
y = \frac{1}{2}x + \frac{3}{2} + \frac{10}{2}
\]
which simplifies to:
\[
y = \frac{1}{2}x + \frac{13}{2}
\]

3. **Convert to standard form.**
The standard form of a linear equation is \(Ax + By = C\). To convert \(y = \frac{1}{2}x + \frac{13}{2}\) to this form:
Multiply through by 2 to eliminate the fraction:
\[
2y = x + 13
\]
Rearranging gives:
\[
-x + 2y = 13
\]
To express it in standard form with \(A\) positive, multiply everything by -1:
\[
x - 2y = -13
\]

So, the equation of the line in standard form that is parallel to \(2x - 4y = 3\) and passes through the point (-3, 5) is:
\[
\boxed{x - 2y = -13}
\]

Answer 3

Apologies for any mistakes. Let's go through the solution step by step once again to find the correct equation of the line through the point \((-3, 5)\) and parallel to the line given by \(2x - 4y = 3\).

### Step 1: Find the slope of the given line
We start with the equation of the line:
\[
2x - 4y = 3
\]

To find the slope, we can rewrite this in slope-intercept form \(y = mx + b\):
\[
-4y = -2x + 3
\]
\[
y = \frac{1}{2}x - \frac{3}{4}
\]

From this, we see that the slope \(m\) is \(\frac{1}{2}\).

### Step 2: Use the point-slope form
Since we want the line that is parallel to the given line, it will also have a slope of \(\frac{1}{2}\). We will use the point-slope form of a line equation, which is:
\[
y - y_1 = m(x - x_1)
\]
Here, \((x_1, y_1) = (-3, 5)\) and \(m = \frac{1}{2}\):
\[
y - 5 = \frac{1}{2}(x + 3)
\]

### Step 3: Expand the point-slope equation
Expanding this equation:
\[
y - 5 = \frac{1}{2}x + \frac{3}{2}
\]
Adding 5 to both sides:
\[
y = \frac{1}{2}x + \frac{3}{2} + 5
\]
Converting 5 to halves:
\[
y = \frac{1}{2}x + \frac{3}{2} + \frac{10}{2}
\]
\[
y = \frac{1}{2}x + \frac{13}{2}
\]

### Step 4: Convert to standard form
To convert \(y = \frac{1}{2}x + \frac{13}{2}\) into standard form \(Ax + By = C\), we can multiply through by 2 to eliminate the fractions:
\[
2y = x + 13
\]
Rearranging gives:
\[
-x + 2y = 13
\]
To write it in standard form with \(A\) positive, we multiply through by \(-1\):
\[
x - 2y = -13
\]

### Final Answer
The correct equation of the line in standard form that passes through \((-3, 5)\) and is parallel to the line \(2x - 4y = 3\) is:
\[
\boxed{x - 2y = -13}
\]

If there is still an issue, please provide any insights on what you believe the mistake is so I can help further. Thank you!