UpStudy Free Solution:

Let's address each part of the question step-by-step.

a. Rewrite the equation \(3x + y - 2 = 0\) in slope-intercept form.

The slope-intercept form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Starting with the given equation:

\[3x + y - 2 = 0\]

First, solve for \(y\):

\[y = - 3x + 2\]

So, the equation in slope-intercept form is:

\[y = - 3x + 2\]

b. Give the slope and y-intercept.

From the equation \(y = - 3x + 2\):

- The slope (\(m\)) is \(- 3\).

- The y-intercept (\(b\)) is \(2\).

c. Use the slope and y-intercept to graph the linear function.

To graph the linear function \(y = - 3x + 2\):

1. Plot the y-intercept: The y-intercept is \(2\), so plot the point \(( 0, 2) \) on the y-axis.

2. Use the slope to find another point: The slope is \(- 3\), which means for every 1 unit increase in \(x\), \(y\) decreases by 3 units. Starting from the y-intercept \(( 0, 2) \):

- Move 1 unit to the right (positive direction of \(x\)).

- Move 3 units down (negative direction of \(y\)).

This gives the point \(( 1, - 1) \).

3. Draw the line: Draw a straight line through the points \(( 0, 2) \) and \(( 1, - 1) \).

The line passes through the points \(( 0, 2) \) and \(( 1, - 1) \), and the slope of the line is \(- 3\).

Supplemental Knowledge

Understanding the slope-intercept form of a linear equation is crucial in algebra. The slope-intercept form is given by:

\[y = mx + b\]

where:

- \(m\) represents the slope of the line, indicating how steep the line is.

- \(b\) represents the y-intercept, which is the point where the line crosses the y-axis.

Key Concepts:

1. Slope (\(m\)):

- The slope indicates the rate of change of \(y\) with respect to \(x\).

- It is calculated as the rise over run (\(\Delta y / \Delta x\)).

- A positive slope means the line ascends from left to right, while a negative slope means it descends.

2. Y-Intercept (\(b\)):

- This is where the line crosses the y-axis (when \(x = 0\)).

- It provides a starting point for graphing the line.

Example Calculation:

Given a linear equation in standard form, such as \(Ax + By + C = 0\), you can convert it to slope-intercept form by solving for \(y\):

1. Start with:

\[Ax + By + C = 0\]

2. Isolate \(y\):

\[By = - Ax - C\]

3. Divide by \(B\):

\[y = - \frac { A} { B} x - \frac { C} { B} \]

Here,

\[m = - \frac { A} { B} \]

and

\[b = - \frac { C} { B} \]

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