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Chapter 4: Problem 45

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

Short Answer

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

Step by step solution

01

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.
02

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 \((-10, 0)\).
03

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 \((0, 4)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥攁ddition, subtraction, multiplication, or division鈥攗ntil 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.

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Most popular questions from this chapter

In 2003 , the owl population in a park was measured to be 340. By 2007, the population was measured again to be \(285 .\) The population changes linearly. Let the input be years since 1990 . a. Find a formula for the owl population, \(P .\) Let the input be years since 2003 . b. What does your model predict the owl population to be in 2012?

Graph the function \(f\) on a domain of [-10,10]\(: f(x)=2,500 x+4,000\).

A clothing business finds there is a linear relationship between the number of shirts, \(n,\) it can sell and the price, \(p,\) it can charge per shirt. In particular, historical data shows that \(1,000\) shirts can can be sold at a price of \(\$ 30,\) while \(3,000\) shirts can be sold at a price of \(\$ 22 .\) Find a linear equation in the form \(p(n)=m n+b\) that gives the price \(p\) they can charge for \(n\) shirts.

For the following exercises, consider this scenario: A town's population has been decreasing at a constant rate. In 2010 the population was \(5,900.5 \mathrm{y} 2012\) the population had dropped \(4,700 .\) Assume this trend continues. Identify the year in which the population will reach 0.

Find the equation of the line that passes through the following points: \((a, 0)\) and \((c, d)\).
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