Determine the x- intercept 'a' and the y-intercept 'b' of the following lines. Sketch each.
`3x+5y-15=0`,

Answer

To determine the x-intercept 'a' and the y-intercept 'b' of the line given by the equation \(3x + 5y - 15 = 0\), we can follow these steps:

### Step 1: Find the x-intercept
To find the x-intercept, we set \(y = 0\) in the equation and solve for \(x\).

\[ 3x + 5(0) - 15 = 0 \]
\[ 3x - 15 = 0 \]
\[ 3x = 15 \]
\[ x = \frac{15}{3} = 5 \]

Thus, the x-intercept \(a\) is \(5\).

### Step 2: Find the y-intercept
To find the y-intercept, we set \(x = 0\) in the equation and solve for \(y\).

\[ 3(0) + 5y - 15 = 0 \]
\[ 5y - 15 = 0 \]
\[ 5y = 15 \]
\[ y = \frac{15}{5} = 3 \]

Thus, the y-intercept \(b\) is \(3\).

### Step 3: Write the intercept form of the equation
The intercept form of the equation of a line can be expressed as:

\[ \frac{x}{a} + \frac{y}{b} = 1 \]

Substituting the values of \(a\) and \(b\):

\[ \frac{x}{5} + \frac{y}{3} = 1 \]

### Step 4: Sketch the line
To sketch the line, we can plot the intercepts on a coordinate plane:

- The x-intercept is at the point \((5, 0)\).
- The y-intercept is at the point \((0, 3)\).

Draw a straight line through these two points, which represents the equation of the line \(3x + 5y - 15 = 0\).

### Summary of Results
- x-intercept \(a = 5\)
- y-intercept \(b = 3\)