Problem 45 - College Algebra

Question

For the following exercises, find the \(x\) - and \(y\) -intercepts of each equation. $$ -2 x+5 y=20 $$

Step-by-Step Solution

Answer

x-intercept: (-10, 0), y-intercept: (0, 4)

Step 1: Identify the Equation

We are given the linear equation \(-2x + 5y = 20\). Our goal is to find the intercepts on the x-axis and y-axis.

Step 2: Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\). \[-2x + 5(0) = 20\]\[-2x = 20\]\[x = \frac{20}{-2}\]\[x = -10\]Thus, the x-intercept is \((-10, 0)\).

Step 3: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\). \[-2(0) + 5y = 20\]\[5y = 20\]\[y = \frac{20}{5}\]\[y = 4\]Thus, the y-intercept is \((0, 4)\).

Key Concepts

Understanding the x-intercept

When you hear about the **x-intercept**, know that it means the point where a line crosses the x-axis on a graph. It's where the value of **y** is zero. To find this intercept in an equation, you simply set the **y** value to zero and solve for **x**.

• This pinpoint will always have the form \( (x, 0) \).
• It helps us to understand where our line sits relative to the x-axis.

In this specific case, for the equation \(-2x + 5y = 20\), substituting \( y = 0 \) results in \( -2x = 20 \). Solving this gives \( x = -10 \), so the x-intercept is \( (-10, 0) \). This means the line crosses the x-axis at **-10**. Use this method to find x-intercepts in any other linear equation on the coordinate plane.

Determining the y-intercept

The **y-intercept** is the point at which a line crosses the y-axis. Here, the **x** value is zero.

1. All y-intercepts can be identified by the form \( (0, y) \).
2. This value shows us the location of a line in relation to the y-axis.

To discover the y-intercept of our example equation, \(-2x + 5y = 20\), set \( x = 0 \). You will have \(5y = 20\), which simplifies to \(y = 4\). Therefore, the y-intercept is \( (0, 4) \), meaning the line will touch the y-axis at **4**. Understanding how to find the y-intercept greatly aids in visualizing how any given line behaves on the graph.

Solving Linear Equations

**Solving equations** is the process of finding what values satisfy the condition set out in an equation. With linear equations, every resolution offers a straight line when graphed, following the standard form \( Ax + By = C \).Here's how we tackle it:

• Substitute known values, like **x = 0** or **y = 0**, to find intercepts.
• Isolate the variable by using basic algebraic operations—addition, subtraction, multiplication, or division—until the variable is alone on one side.

In our example with \(-2x + 5y = 20\), steps were neatly executed by setting **x** or **y** to zero and applying simple rearrangements. Effective solving of these equations lets you find critical points like intercepts, making it easier to sketch the line on a graph.

Getting Familiar with the Coordinate Plane

The **coordinate plane** is a two-dimensional space defined by two numbers, the x-coordinate and the y-coordinate. Here is what makes it special:

• The plane is divided by a horizontal line called the **x-axis** and a vertical line known as the **y-axis**.
• Intersections of these axes are called **origin**, represented as \( (0,0) \).
• Different points on this plane are identified based on their \( (x,y) \) coordinates, which precisely position them.

In problems dealing with linear equations like \(-2x + 5y = 20\), knowing how to navigate this plane helps plot lines by placing intercepts accurately. Understanding these coordinates is vital because they translate abstract numbers into visually interpretable graphs, making concepts come alive with geometry.