Problem 56 - Algebra for College Students

Question

Find the \(x\) - and \(y\) -intercepts for each line and use them to graph the line. $$y=-3 x+6$$

Step-by-Step Solution

Answer

The x-intercept is (2, 0) and the y-intercept is (0, 6).

Step 1 - Find the y-intercept

To find the y-intercept, set \(x = 0\) and solve for \(y\). Substitute 0 for \(x\) in the equation: \[ y = -3(0) + 6 \]Thus, \( y = 6 \). The y-intercept is \((0, 6)\).

Step 2 - Find the x-intercept

To find the x-intercept, set \(y = 0\) and solve for \(x\). Substitute 0 for \(y\) in the equation: \[ 0 = -3x + 6 \]Solving for \(x\): \[ 3x = 6 \] \[ x = 2 \]. The x-intercept is \((2, 0)\).

Step 3 - Plot the intercepts

Plot the points \((0, 6)\) and \((2, 0)\) on the coordinate plane. These are the intercepts found in Steps 1 and 2.

Step 4 - Draw the line

Draw a straight line passing through the points \((0, 6)\) and \((2, 0)\) to graph the line.

Key Concepts

intercepts

When graphing linear equations, intercepts are key points where the line crosses the axes. The two main types of intercepts are the x-intercept and the y-intercept.
The y-intercept is the point where the line crosses the y-axis. To find it, set the value of x to 0 and solve the equation for y. For example, in the equation you provided, for the y-intercept, we set x to 0:
\[ y = -3(0) + 6 = 6 \]This means our y-intercept is at the point (0, 6).
Similarly, the x-intercept is where the line crosses the x-axis. To find this, set the value of y to 0 and solve for x:\[ 0 = -3x + 6 \]Solving for x, we get x = 2, giving us the x-intercept at (2, 0).
Finding intercepts helps in plotting the line easily. We'll look into plotting the points on the coordinate plane next.

coordinate plane

The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). These two axes intersect at the origin, denoted as (0, 0).
Each point on the plane is represented as an ordered pair (x, y). The first number is the x-coordinate, which shows the distance to the right or left of the y-axis. The second number is the y-coordinate, showing the distance above or below the x-axis.
To graph the equation \( y = -3x + 6 \), we plotted the intercepts: (0, 6) and (2, 0). Here's how you do it:

• Step 1: Mark the y-intercept (0, 6) on the y-axis.
• Step 2: Mark the x-intercept (2, 0) on the x-axis.
• Step 3: Draw a straight line through both points to represent the equation.

By graphing the intercepts and drawing the line through them, we get a visual representation of the linear equation.

solving equations

Solving equations is a fundamental skill in algebra. It involves finding the value of the variable that makes the equation true. In the given problem \( y = -3x + 6 \), we solved for the intercepts by setting one variable to zero and solving for the other. Let's break this down:

• To find the y-intercept: Set x to 0 and solve for y:
\[ y = -3(0) + 6 = 6 \]So, the y-intercept is (0, 6).
• To find the x-intercept: Set y to 0 and solve for x:
\[ 0 = -3x + 6 \]Rearranging and solving for x:\[ 3x = 6 \]\[ x = 2 \]So, the x-intercept is (2, 0).

By finding these intercepts and solving the equation, we are able to plot the line on a graph, helping us visualize the relationship between the variables. Mastering these steps is crucial for understanding more complex algebraic concepts.