Problem 53 - Algebra for College Students

Question

Find the \(x\) - and \(y\) -intercepts for each line and use them to graph the line. $$2 x-3 y=60$$

Step-by-Step Solution

Answer

The x-intercept is at (30, 0) and the y-intercept is at (0, -20).

Step 1 - Identify the Equation Form

The given equation is in the form of a linear equation: \[ 2x - 3y = 60 \]

Step 2 - Find the x-intercept

To find the x-intercept, set y to 0 in the equation: \[ 2x - 3(0) = 60 \] This simplifies to \[ 2x = 60 \] Solving for x gives \[ x = 30 \] So, the x-intercept is at the point (30, 0).

Step 3 - Find the y-intercept

To find the y-intercept, set x to 0 in the equation: \[ 2(0) - 3y = 60 \] This simplifies to \[ -3y = 60 \] Solving for y gives \[ y = -20 \] So, the y-intercept is at the point (0, -20).

Step 4 - Plot the Intercepts

Plot the x-intercept (30, 0) and the y-intercept (0, -20) on the graph.

Step 5 - Draw the Line

Draw a straight line through the points (30, 0) and (0, -20). This line represents the equation \[ 2x - 3y = 60 \].

Key Concepts

x-intercept

The x-intercept of a line is the point where it crosses the x-axis. To find this, you set the y-value to 0 in the equation and solve for x. As shown in the exercise, for the equation \[ 2x - 3y = 60 \], setting y to 0 simplifies the equation to \[ 2x = 60 \]. Solving for x, we get \[ x = 30 \]. Therefore, the x-intercept is the point (30, 0). Remember, the x-intercept is always where the graph meets the x-axis, meaning the y-coordinate is 0 at this point.

y-intercept

The y-intercept is the point where the line crosses the y-axis. To find this, you set the x-value to 0 in the equation and solve for y. For the given equation \[ 2x - 3y = 60 \], setting x to 0 simplifies it to \[ -3y = 60 \]. Solving for y, we get \[ y = -20 \]. Hence, the y-intercept is the point (0, -20). The y-intercept is particularly important because it's the starting point when graphing a line on the Cartesian plane, and it's where the graph meets the y-axis.

plotting points

Plotting points involves marking specific locations on a graph that correspond to coordinates \((x, y)\). In our case, these coordinates are the x-intercept (30, 0) and the y-intercept (0, -20). To plot these points:

• Locate the x-intercept, (30, 0), by moving 30 units right along the x-axis.
• Locate the y-intercept, (0, -20), by moving 20 units down along the y-axis.

Once these points are plotted, you can draw a line through them, which represents the equation. Plotting points accurately ensures the correct representation of the line on the graph.

linear equations

Linear equations are algebraic expressions that represent straight lines when graphed. The general form of a linear equation is \[ax + by = c\], where a, b, and c are constants. Solving a linear equation often involves finding intercepts or other points. For the given equation \[ 2x - 3y = 60 \], we:

• Set y to 0 to find the x-intercept.
• Set x to 0 to find the y-intercept.
• Use these intercepts to plot the line on a graph.

Linear equations are fundamental in algebra and are widely used to model real-world situations where relationships between two variables are linear.