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Chapter 3: Problem 18

Find the \(x\) - and \(y\) -intercepts for the graph of each equation. $$ x-y=7 $$

Short Answer

Expert verified
The x-intercept is (7, 0) and the y-intercept is (0, -7).

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. The given equation is \(x - y = 7\). Set y to 0: \(x - 0 = 7\). Thus, \(x = 7\). The x-intercept is (7, 0).
02

Find the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y. The given equation is \(x - y = 7\). Set x to 0: \(0 - y = 7\). Thus, \(-y = 7\), so \(y = -7\). The y-intercept is (0, -7).

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Key Concepts

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

find x-intercept
Finding the x-intercept of a line in a graph is an important skill in algebra. This involves locating the point where the line crosses the x-axis. To find the x-intercept, follow these simple steps:
  • Set y to 0 in the equation.
  • Solve for x.
In the example from the exercise, the given equation was
\ x - y = 7

Let's find the x-intercept by setting y to 0:
\( x - 0 = 7 \) \ Therefore, \( x = 7 \).

This calculation tells us that when y is 0, x is 7. So, the coordinates of the x-intercept are (7, 0). This means the line crosses the x-axis at the point (7, 0).
  • Understanding intercepts helps to quickly sketch the graph of the equation and analyze its essential features.

    • This concept is not just limited to one type of equation but extends to various forms and complexities of linear equations.
    find y-intercept
    Just like finding the x-intercept, finding the y-intercept is crucial for graph drawing. It involves locating the point where the line crosses the y-axis. Here's the step-by-step process:
    • Set x to 0 in the equation.
    • Solve for y.
    Returning to the exercise example, the equation is:
    \ x - y = 7

    Let's find the y-intercept by setting x to 0:
    \( 0 - y = 7 \) \ This simplifies to \( - y = 7 \),

    which means \( y = -7 \).

    This tells us that when x is 0, y is -7. So, the coordinates of the y-intercept are (0, -7). This means the line crosses the y-axis at the point (0, -7).

    • Mastering the concept of intercepts makes it easier to graph lines accurately and understand their behavior quickly.
    • Intercepts can also indicate the solutions of equations when plotted graphically.
    linear equations
    Linear equations are foundational in algebra and are essential in many areas of mathematics and real-world applications. A linear equation typically takes the form
    \ax + by = c

    Where a, b, and c are constants. The graph of a linear equation is a straight line.
    Here are some characteristics and tips to understand linear equations better:
    • They have a constant rate of change, known as the slope.
    • The graph is always a straight line, either rising, falling, or remaining horizontal or vertical depending on the equation.
    • In the standard form, \(ax + by = c\), the coefficients (a, b) and constant (c) can help determine the slope and intercept of the equation.
    • The x- and y-intercepts are specific points where the line crosses the axes, found by setting y=0 for x-intercept, and setting x=0 for y-intercept, as explained previously.

    Knowing how to manipulate and graph these equations simplifies understanding more complex algebraic concepts and enhances problem-solving abilities.

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    Most popular questions from this chapter

    \(9 x-2 y=0\) \(-18 x+4 y=3\)

    Solve each problem. The weight \(y\) (in pounds) of a man taller than 60 in. can be approximated by the linear equation $$ y=5.5 x-220 $$ where \(x\) is the height of the man in inches. (a) Use the equation to approximate the weights of men whose heights are 62 in., 66 in., and 72 in. (b) Write the information from part (a) as three ordered pairs. (c) Graph the equation for \(x \geq 62\), using the data from part (b). (d) Use the graph to estimate the height of a man who weighs 155 lb. Then use the equation to find the height of this man to the nearest inch.

    Describe what the graph of each linear equation will look like in the coordinate plane. (Hint: Rewrite the equation if necessary so that it is in a more recognizable form.) $$ x-10=1 $$

    Each table of values gives several points that lie on a line. (a) Use any two of the ordered pairs to find the slope of the line. (b) What is the x-intercept of the line? The y-intercept? (c) Graph the line. $$ \begin{array}{r|r} \hline x & y \\ \hline-4 & 0 \\ \hline-2 & 2 \\ \hline 0 & 4 \\ \hline 1 & 5 \end{array} $$

    Write an equation of the line satisfying the given conditions. Give the final answer in slope-intercept form. (Hint: Recall the relationships among slopes of parallel and perpendicular lines.) Passes through (2,3) parallel to \(4 x-y=-2\)
    See all solutions

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