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Chapter 4: Problem 156

Find the intercepts for each equation. \(x-y=1\)

Short Answer

Expert verified
The intercepts are (1, 0) and (0, -1).

Step by step solution

01

- Find the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x. Start with the original equation:\[ x - y = 1 \]When y is 0:\[ x - 0 = 1 \]\[ x = 1 \]Therefore, the x-intercept is (1, 0).
02

- Find the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y. Start with the original equation:\[ x - y = 1 \]When x is 0:\[ 0 - y = 1 \]\[ -y = 1 \]Multiply both sides by -1 to solve for y:\[ y = -1 \]Therefore, the y-intercept is (0, -1).

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Key Concepts

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

x-intercept
To find the x-intercept of a linear equation, you need to set the value of y to zero. This will show where the line crosses the x-axis.
Start with your given equation and substitute y with 0:
For example, with the equation \(x - y = 1\), set \(y = 0\):
\(x - 0 = 1\)
Then, simplify and solve for x:
\(x = 1\)
Thus, your x-intercept is (1, 0). This is the point where the graph touches the x-axis at x=1 while y remains 0.
This concept is very useful in graphing because it helps in drawing the first point of the line on the coordinate plane.
y-intercept
To find the y-intercept of a linear equation, set the value of x to zero. This will show where the line crosses the y-axis.
Take the original equation and substitute x with 0:
For example, with the equation \(x - y = 1\), set \(x = 0\):
\(0 - y = 1\)
Then, simplify and solve for y:
\(-y = 1\)
Multiply both sides by -1 to isolate y:
\(y = -1\)
Thus, your y-intercept is (0, -1). This is the point where the graph touches the y-axis at y=-1 while x remains 0.
Finding the y-intercept helps identify where the line begins on the vertical axis.
solving linear equations
Solving linear equations is a fundamental skill in math. Linear equations form straight lines when graphed.
To solve them, you can use various methods such as:
  • Substitution: Replacing variables with values to simplify the equation
  • Elimination: Combining equations to cancel out a variable
  • Graphing: Plotting the equation on a graph to find intercepts
Consider the equation: \(x - y = 1\)
When we set y to 0 to find the x-intercept, we solved the simplified equation \(x = 1\).
When we set x to 0 to find the y-intercept, we simplified to \(-y = 1\) and then \(y = -1\).
These methods help you understand how each variable affects the equation and find precise points on the graph where the line crosses axes.
Understanding how to solve linear equations is key to mastering algebra and many real-world applications.

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Most popular questions from this chapter

Find the equation of a line with given slope and containing the given point. Write the equation in slope-intercept form. Horizontal line containing (4,-8)

Find the equation of a line containing the given points. Write the equation in slope-intercept form. (3,5) and (-7,5)

Cherie works in retail and her weekly salary includes commission for the amount she sells. The equation \(S=400+0.15 c\) models the relation between her weekly salary, \(S\), in dollars and the amount of her sales, \(c\), in dollars. (a) Find Cherie's salary for a week when her sales were 0 (b) Find Cherie's salary for a week when her sales were 3600 . (c) Interpret the slope and S-intercept of the equation. (d) Graph the equation.

Graph using the intercepts. \(-x+4 y=8\)

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