Problem 57 - College Algebra with Modeling and Visualization

Question

Determine the \(x\) - and \(y\) -intercepts on the graph of the equation. Graph the equation. \(4 x-5 y=20\)

Step-by-Step Solution

Answer

The x-intercept is (5, 0) and the y-intercept is (0, -4); graph using these points.

Step 1: Find the x-intercept

To find the x-intercept, set the value of \(y\) to 0 and solve for \(x\). Thus, the equation becomes \(4x - 5(0) = 20\). Simplifying gives you \(4x = 20\). Divide both sides by 4 to solve for \(x\), giving \(x = 5\). So, the x-intercept is (5, 0).

Step 2: Find the y-intercept

To find the y-intercept, set the value of \(x\) to 0 and solve for \(y\). Thus, the equation becomes \(4(0) - 5y = 20\). Simplifying gives you \(-5y = 20\). Divide both sides by -5 to solve for \(y\), giving \(y = -4\). So, the y-intercept is (0, -4).

Step 3: Graphing the Equation

Now that we know the intercepts, we can graph the line using the points (5, 0) and (0, -4). Plot the points on the graph, with the x-intercept at (5, 0) and the y-intercept at (0, -4). Draw a straight line through these points, extending it across the graph.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a fundamental skill in algebra that allows us to visually represent relationships between variables. A linear equation is typically in the form of ax + by = c, where x and y are variables and a, b, and c are constants.

To graph a linear equation, you identify its key characteristics, like intercepts and slope, to plot it on a graph. The process generally involves the following steps:

• Determine the x-intercept and y-intercept.
• Plot these intercepts on the Cartesian plane.
• Draw a straight line that passes through the points.

These steps are crucial because the intercepts help define the line's position and direction. The line represents all solutions to the equation. This approach makes solving equations and understanding algebraic relationships easier.

With a clear graph, you can examine how the variables interact. It also helps in predicting values and understanding real-world scenarios modeled by these equations. Ensuring accuracy while plotting leads to a better understanding of the equation's behavior.

x-intercept

The x-intercept is a vital concept that refers to the point where the graph crosses the x-axis. To find the x-intercept, set the value of y to zero and solve for x. This is because the x-axis itself is defined by y = 0.

In our example equation, 4x - 5y = 20, you simplify by plugging in y = 0:

- Substitute: 4x - 5(0) = 20, simplifying to 4x = 20. - Solve for x by dividing both sides by 4, yielding x = 5.

Thus, the x-intercept is at the point (5, 0).

A few points to remember about x-intercepts:

• The x-intercept is always written as (a number, 0).
• If an equation has no x-intercept, the graph doesn't cross the x-axis.
• X-intercepts are useful in predicting when a scenario modeled by the equation might start or stop.

This intercept helps us understand at what point variable x will exist independently, without the influence of y – an essential feature when graphing linear equations.

y-intercept

The y-intercept signifies the point where the line crosses the y-axis, occurring when x equals zero. To find it, you substitute x = 0 into the equation and solve for y.

Applying this to 4x - 5y = 20, replace x with 0:

- Substitute: 4(0) - 5y = 20, which simplifies to -5y = 20. - Solve for y by dividing by -5, resulting in y = -4.

So, the y-intercept is at (0, -4).

Important aspects of y-intercepts include:

• The y-intercept is noted as (0, a number).
• In cases where there's no y-intercept, the graph doesn't touch or cross the y-axis.
• Y-intercepts are often initial starting points in models or represent a base value.

This intercept is essential for understanding the point at which every potential value of x produces a zero, marking a significant feature of the linear graph.