Problem 7 - Thinking Mathematically

Question

Use the \(x\) - and \(y\)-intercepts to graph each linear equation. \(5 x=3 y-15\)

Step-by-Step Solution

Answer

The x and y intercepts of the linear equation \(5 x=3 y-15\) are -3 and 5 respectively. The points (-3,0) and (0,5) are plotted on the graph and a line is drawn through them.

Step 1: Find the x intercept

To find the x intercept, we'll make y = 0 in the equation \(5x = 3y - 15\). Substituting values, we have \(5x = 3(0) - 15 \Rightarrow 5x = -15 \Rightarrow x = -15 / 5 \Rightarrow x = -3\). So, the x intercept is -3.

Step 2: Find the y intercept

To find the y intercept, make x = 0 in the equation. Hence, \(5(0) = 3y - 15 \Rightarrow 0 = 3y -15 \Rightarrow 15 = 3y \Rightarrow y = 15/3 \Rightarrow y = 5\). So, the y intercept is 5.

Step 3: Graph the equation

Now, plot the x intercept at (-3, 0) and the y intercept at (0, 5) on a graph. Draw a straight line through these points to create a visual representation of the linear equation.

Key Concepts

Understanding the X-Intercept

The x-intercept is where a graph crosses the x-axis. This point happens when the y-value is zero, because you’re not moving up or down on the graph. To find the x-intercept of a linear equation, you need to set the y-variable to 0 and solve for x.
For the equation given, which is initially in the form of \(5x = 3y - 15\), you'd place zero in place of y:

• Set \(y = 0\) \(\Rightarrow 5x = 3(0) - 15\)
• Solve for \(x\): \(5x = -15 \Rightarrow x = \frac{-15}{5} = -3\)

Thus, the x-intercept is the point \((-3, 0)\). This point is crucial because it helps anchor the line on the graph.

Exploring the Y-Intercept

In contrast to the x-intercept, the y-intercept occurs where the graph crosses the y-axis. Here, the x-value is zero, hence you move up or down along the y-axis. To determine the y-intercept, substitute 0 for x in the equation and solve for y.
For the given equation \(5x = 3y - 15\), insert zero for x:

• Set \(x = 0\) \(\Rightarrow 5(0) = 3y - 15\)
• Solve for \(y\): \(0 = 3y - 15 \Rightarrow 3y = 15 \Rightarrow y = \frac{15}{3} = 5\)

Therefore, the y-intercept is the point \((0, 5)\). This y-intercept provides another anchor point essential for graphing the line.

Defining a Linear Equation

Linear equations are expressions that represent straight lines when graphed. They typically come in the form \(Ax + By = C\), where A, B, and C are constants.
In the context of this exercise, the equation provided is \(5x = 3y - 15\), which can be rearranged to the standard form \(5x - 3y = -15\). Every linear equation corresponds to a line that has constant rate of change and doesn’t curve.
Key characteristics include:

• Consistent rate of change (slope)
• Straight line
• Two intercepts: x-intercept and y-intercept

Understanding linear equations helps in predicting values and interpreting data relationships on a graph.

Graphing with Intercepts

Graphing is a visual way to represent equations and explore the relationships between variables. Using the intercepts is a convenient method for graphing linear equations, especially when coordinates are easy to compute.

The process involves the following steps:

• Identify the x-intercept, which is the point on the graph where the line crosses the x-axis, here at \((-3, 0)\).
• Identify the y-intercept, which is the point where the line crosses the y-axis, here at \((0, 5)\).
• Plot these intercepts on a coordinate plane.
• Draw a straight line through the points.

The line represents all solutions to the linear equation \(5x = 3y - 15\). This technique simplifies the graphing process and helps interpret how changes in x and y affect each other in the equation.