Problem 43 - Algebra 1: Concepts and Skills

Question

Find the x-intercepts and the y-intercepts of the line. Graph the equation. Label the points where the line crosses the axes. $$ y=-3 x+9 $$

Step-by-Step Solution

Answer

The x-intercept is at (3, 0) and the y-intercept is at (0, 9).

Step 1: Finding the X-intercept

Set y = 0 in the equation to find the x-coordinate when the line crosses the x-axis:0 = -3x + 9 First isolate 3x on one side of the equation:3x = 9 Now, divide both sides by 3 to solve for x:x = 9 / 3 Therefore, the x-intercept is \(x = 3\). The point is (3, 0).

Step 2: Finding the Y-intercept

Set x = 0 in the equation to find the y-coordinate when the line crosses the y-axis:y = -3(0) + 9 y = 9 Therefore, the y-intercept is \(y = 9\). The point is (0, 9).

Step 3: Graphing the Equation

First plot the x-intercept at (3, 0) and the y-intercept at (0, 9). Then connect the two points to form a line. Lastly, label the intercept points on the graph.

Key Concepts

X-intercepts of a Line

Understanding the x-intercept of a line is fundamental in graphing linear equations. An x-intercept is a point on the graph where the line crosses the x-axis. At this point, the value of y is zero. To find the x-intercept, you simply set the y-coordinate to zero and solve for x.

Following the given exercise, the equation provided is \( y = -3x + 9 \). To find the x-intercept, we replace y with 0, yielding \( 0 = -3x + 9 \). By solving for x, you find that the x-intercept is at \( x = 3 \), corresponding to the point (3, 0) on the coordinate plane. This method helps clarify where the line will touch the x-axis, making it easier for you to graph the line accurately.

It is crucial for students to understand that every linear equation will have at most one x-intercept, which can be found using this simple procedure.

Y-intercepts of a Line

Similarly, the y-intercept of a line is where the line crosses the y-axis. At this point, the x-coordinate is zero. Finding the y-intercept requires setting the value of x to zero in the equation and solving for y.

Continuing with our exercise, the equation is still \( y = -3x + 9 \). By setting \( x = 0 \), we obtain \( y = -3(0) + 9 \), which simplifies to \( y = 9 \). This indicates that the y-intercept is at the point (0, 9). Knowing the y-intercept provides a starting point for graphing the line and is equally important for understanding the basics of the graph's orientation relative to the y-axis.

Remember that a line can have only one y-intercept, and this point is essential to graph linear equations accurately and to understand their graphical representations.

Solving Linear Equations

Solving linear equations is the cornerstone of understanding how to graph them. A linear equation represents a straight line and can be written in the form \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. To solve these equations, you perform algebraic operations to isolate the variable you are solving for, whether it's x or y.

When graphing, like in our exercise, finding both the x and y intercepts simplifies the process. Once you have these two points, (3, 0) and (0, 9) from our previous sections, you simply plot them on a graph and draw a line through them. This visual representation helps students better understand the concept of a function and the relationship between variables within an equation.

Clear, step-by-step approaches in solving for x and y intercepts not only aid in graphing but also build a student's foundational algebra skills that are applicable in more complex mathematical contexts.