Question

Find the slope and y-intercept of each line. Graph the line. $$ 3 x+2 y=6 $$

Short answer

Slope: \( -\frac{3}{2} \), y-intercept: 3.

Step-by-step solution

- Rewrite the equation in slope-intercept form

To find the slope and y-intercept, convert the given equation into slope-intercept form \(y = mx + b\). Start by isolating y: \ 3x + 2y = 6 \ becomes \ 2y = -3x + 6 \ and then \ y = -\frac{3}{2}x + 3 \.

- Identify the slope (m) and y-intercept (b)

Now that the equation is in the form \(y = mx + b\), we can identify the slope (m) and the y-intercept (b). Here, the slope \( m = -\frac{3}{2} \) and the y-intercept \( b = 3 \).

- Plot the y-intercept on the graph

Start by plotting the y-intercept (b) on the y-axis. The y-intercept is 3, so place a point at \( (0, 3) \).

- Use the slope to find another point

The slope \( -\frac{3}{2} \) tells us to go down 3 units and to the right 2 units from the y-intercept to locate another point. Thus, from \( (0, 3) \), move to \( (2, 0) \). Plot this point.

- Draw the line

Finally, draw a straight line through the points \( (0, 3) \) and \( (2, 0) \), extending it in both directions to represent the entire line.

Key concepts

slope-intercept form

The slope-intercept form is a common way to represent linear equations. This form is written as \(y = mx + b\). In this equation, **m** represents the slope of the line, while **b** is the y-intercept. The **slope** indicates how steep the line is, and the **y-intercept** is the point where the line crosses the y-axis.
To convert an equation into slope-intercept form, you isolate **y** on one side of the equation. This makes it easy to identify both the slope and the y-intercept directly from the equation.

graphing linear equations

Graphing linear equations involves plotting points on a coordinate plane and drawing a line through them. First, identify the y-intercept from the equation and plot this point on the y-axis. Next, use the slope to find additional points. The slope, given as a fraction, dictates how to move from one point to another. For example, a slope of \(-3/2\)\ tells you to go down 3 units and to the right 2 units to find the next point. Plot these points and draw a line through them to represent the linear equation. Line equations extend infinitely, so draw the line through the points and beyond the edges of the graph.

identifying slope and y-intercept

To identify the slope and y-intercept of a linear equation, the equation must first be in slope-intercept form \ (y = mx + b) \. With the equation \(y = -\frac{3}{2}x + 3\), note that:

• The slope **m** is \(-\frac{3}{2}\).
• The y-intercept **b** is 3.

The slope shows the rate of change; it indicates that for every 2 units you move right (positive direction), you move down 3 units (negative direction). The y-intercept tells you the starting point of the line on the y-axis.
Identifying these key features helps to quickly sketch the graph or understand the behavior of the line without plotting multiple points.

linear equations

Linear equations are mathematical expressions that describe a straight line on a graph. They are usually written in forms like slope-intercept form (\( y = mx + b \)) or standard form (\( Ax + By = C \)).
These equations have the highest degree of 1 and involve two variables, typically **x** and **y**. Solving a linear equation means finding the value of **y** for any given **x**. Linear equations are foundational in algebra and are vital for understanding more complex mathematical concepts.
In real life, linear equations model relationships where one quantity depends on another in a constant way, such as speed, cost, or rate.

Understanding linear equations opens the door to analyzing various real-world situations through mathematical expressions.