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Chapter 3: Problem 67

Find all intercepts for each line. Some of these lines have only one intercept. $$x+y=50$$

Short Answer

Expert verified
The intercepts are (50, 0) and (0, 50).

Step by step solution

01

Understand the Equation

The given equation is \(x + y = 50\). This is a linear equation in two variables, which represents a straight line.
02

Find the x-Intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\). \[ x + 0 = 50 \] Thus, \(x = 50\). So, the x-intercept is at the point \((50, 0)\).
03

Find the y-Intercept

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\). \[ 0 + y = 50 \] Thus, \(y = 50\). So, the y-intercept is at the point \((0, 50)\).
04

Summarize Intercepts

The line intersects the x-axis at (50, 0) and the y-axis at (0, 50).

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Key Concepts

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

linear equations
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. In our equation, \(x + y = 50\), it represents a straight line on a graph. Linear equations in this form make it easy to find intercepts, as the relationship between the variables is straightforward.
Linear equations are crucial because they describe simple relationships between variables and can be used in various fields such as physics, economics, and engineering.
x-intercept
The x-intercept is where a line crosses the x-axis. To find it, set the value of y to zero in your equation. For our equation \(x + y = 50\), we set y to 0, so it becomes \(x + 0 = 50\). Solving this, we get \(x = 50\). This means the line crosses the x-axis at (50, 0).
Understanding the x-intercept helps in plotting the line accurately on a graph. It gives one of the two key points needed to visualize your linear equation.
y-intercept
The y-intercept is where the line crosses the y-axis. To find the y-intercept, set the value of x to zero. Using our equation \(x + y = 50\), we set x to 0, which simplifies the equation to \(0 + y = 50\). Solving for y, we get \(y = 50\). Hence, the y-intercept is at (0, 50).
The y-intercept is particularly important because, along with the x-intercept, it allows for easy plotting of the line. These two intercepts define the line completely in a 2D space.
solving equations
Solving equations involves finding the values of the variables that make the equation true. For linear equations, this often means isolating the variable of interest. We did this in the steps for finding both x- and y-intercepts.
For the x-intercept, setting y to 0 and solving for x showed where the line meets the x-axis. Similarly, for the y-intercept, setting x to 0 helped us find where the line intersects the y-axis. These techniques are fundamental for solving linear equations in various contexts.
Remember, the methods used here can be applied to any linear equation, helping you find intersections and better understand the behavior of lines on a graph.

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Most popular questions from this chapter

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