Find the slope intercept form and intercept form of a line 3x + 4y = 5.
(a)-3/4
(b)1/2
(c)5/6
(d)3/4

Answer

To find the slope-intercept form and intercept form of the line given by the equation \(3x + 4y = 5\), we will follow these steps:

### Step 1: Rewrite the equation in slope-intercept form
The slope-intercept form of a line is given by the equation:
\[ y = mx + c \]
where \(m\) is the slope and \(c\) is the y-intercept.

Starting with the equation:
\[ 3x + 4y = 5 \]

We need to solve for \(y\):
1. Subtract \(3x\) from both sides:
\[ 4y = 5 - 3x \]
2. Divide every term by \(4\):
\[ y = \frac{5}{4} - \frac{3}{4}x \]

Now, we can rearrange it to match the slope-intercept form:
\[ y = -\frac{3}{4}x + \frac{5}{4} \]

From this, we can identify:
- Slope \(m = -\frac{3}{4}\)
- Y-intercept \(c = \frac{5}{4}\)

### Step 2: Write the intercept form
The intercept form of a line is given by:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
where \(a\) is the x-intercept and \(b\) is the y-intercept.

To convert \(3x + 4y = 5\) into intercept form, we can rewrite it as follows:
1. Divide the entire equation by \(5\):
\[ \frac{3x}{5} + \frac{4y}{5} = 1 \]

Now, we can identify:
- The x-intercept \(a = \frac{5}{3}\) (when \(y = 0\))
- The y-intercept \(b = \frac{5}{4}\) (when \(x = 0\))

Thus, the intercept form is:
\[ \frac{x}{\frac{5}{3}} + \frac{y}{\frac{5}{4}} = 1 \]

### Summary of Results
- **Slope-Intercept Form**: \(y = -\frac{3}{4}x + \frac{5}{4}\)
- **Intercept Form**: \(\frac{x}{\frac{5}{3}} + \frac{y}{\frac{5}{4}} = 1\)