The problem from the image is asking for two tasks:

Part (a)

• Find the equation of the line that passes through the points $(-1, -3)$ and $(1, -1)$.
• Express the answer in standard form.

Part (b)

• Use the standard form of the equation from part (a) to find the x- and y-intercepts.

Let's first solve part (a) by determining the equation of the line.

Step 1: Calculate the slope

The slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Using the points $(-1, -3)$ and $(1, -1)$: $m = \frac{-1 - (-3)}{1 - (-1)} = \frac{-1 + 3}{1 + 1} = \frac{2}{2} = 1$

Step 2: Use the point-slope form

The point-slope form of the equation of a line is: $y - y_1 = m(x - x_1)$ Substitute $m = 1$ and the point $(-1, -3)$:

$y - (-3) = 1(x - (-1))$ $y + 3 = 1(x + 1)$ $y + 3 = x + 1$

Step 3: Rearrange to standard form

Standard form is $Ax + By = C$, where $A$, $B$, and $C$ are integers. Start by moving all terms to one side:

$y = x + 1 - 3$ $y = x - 2$

To convert to standard form, subtract $x$ from both sides: $x - y = 2$

This is the equation of the line in standard form.

Part (b)

Now, let's find the x-intercept and y-intercept from the standard form equation $x - y = 2$.

• To find the x-intercept, set $y = 0$: $x - 0 = 2 \implies x = 2$ Thus, the x-intercept is $(2, 0)$.

• To find the y-intercept, set $x = 0$: $0 - y = 2 \implies -y = 2 \implies y = -2$ Thus, the y-intercept is $(0, -2)$.

Summary

• The equation of the line in standard form is $x - y = 2$.
• The x-intercept is $(2, 0)$.
• The y-intercept is $(0, -2)$.

Would you like further details or explanations?

5 Related Questions:

1. How would you solve if one of the points involved a fraction?
2. How can the slope be used to identify parallel and perpendicular lines?
3. What is the significance of the standard form of a linear equation in graphing?
4. How would you find the equation of a line if only the slope and one point were given?
5. Can you generalize the relationship between the x- and y-intercepts for any linear equation?

Tip:

When working with linear equations, always double-check your slope and intercepts, as small errors in calculation can affect the final result!