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Chapter 1: Problem 23

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. $$y=5 x-6$$

Short Answer

Expert verified
The \(x\)-intercept is \(\frac{6}{5}\) and the \(y\)-intercept is \(-6\).

Step by step solution

01

Find the \(x\)-intercept

To find the \(x\)-intercept, it's needed to set \(y\) equal to \(0\) in the equation and solve for \(x\). Thus, the equation becomes \(0=5x-6\)
02

Solve the equation for \(x\)

Solving \(0=5x-6\) for \(x\) leads to \(x=\frac{6}{5}\). This is the \(x\)-intercept of the graph.
03

Find the \(y\)-intercept

To find the \(y\)-intercept, the \(x\) in the equation should be set equal to \(0\) and solve for \(y\). Thus, the equation becomes \(y=5*0-6\)
04

Solve the equation for \(y\)

Solving \(y=5*0-6\) for \(y\) we get \(y=-6\). This is the \(y\)-intercept of the graph.

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Key Concepts

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

X-Intercept
The x-intercept of an equation is the point where the graph intersects the x-axis. At this point, the y-value is always zero.
To find the x-intercept in a linear equation, you simply set y to zero and solve the equation for x.
In the given equation, \( y = 5x - 6 \), you replace \( y \) with zero, resulting in the equation \( 0 = 5x - 6 \).
This needs to be solved for x to find the x-intercept.
  • Step 1: Set y to zero, given as \( 0 = 5x - 6 \).
  • Step 2: Add 6 to both sides to get \( 5x = 6 \).
  • Step 3: Divide both sides by 5, which simplifies to \( x = \frac{6}{5} \).
The x-intercept of this graph is \( \left( \frac{6}{5}, 0 \right) \). This is an essential part of understanding graphs as it shows where they cross the x-axis.
Y-Intercept
The y-intercept is the point where the graph crosses the y-axis. In this case, x is always equal to zero.
To find the y-intercept, set x equal to zero in the equation and solve for y.
Using the linear equation \( y = 5x - 6 \), substitute x with zero, resulting in \( y = 5(0) - 6 \).
Once done, calculate the remaining equation to find y.
  • Step 1: Substitute x with zero, hence \( y = 5(0) - 6 \).
  • Step 2: Calculate \( y = 0 - 6 \).
  • Step 3: Simplify to find \( y = -6 \).
Thus, the y-intercept of the graph is \( (0, -6) \). Understanding the y-intercept helps determine where the line crosses the y-axis, a vital concept in graphing linear equations effectively.
Linear Equations
Linear equations are a type of equation involving two variables that produce a straight line when graphed. They are often written in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Understanding the components of a linear equation:
  • Slope (m): Indicates the steepness of the line and direction. A positive slope means the line rises, while a negative slope means it falls.
  • Y-intercept (b): The point where the line crosses the y-axis.
For the equation \( y = 5x - 6 \),
  • The slope \( m \) is 5, meaning the line rises sharply.
  • The y-intercept \( b \) is -6, which we have already found when x is zero.
These linear equations give a straightforward way to depict relationships between two quantities. By knowing how to identify and calculate intercepts and understanding the slope, you can graph and analyze lines quickly and accurately.

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Most popular questions from this chapter

Determine whether the function has an inverse function. If it does, then find the inverse function. $$f(x)=\left\\{\begin{array}{ll}x+3, & x<0 \\\6-x, & x \geq 0\end{array}\right.$$

(a) Given a function \(f,\) prove that \(g(x)\) is even and \(h(x)\) is odd, where \(g(x)=\frac{1}{2}[f(x)+f(-x)]\) and \(h(x)=\frac{1}{2}[f(x)-f(-x)].\) (b) Use the result of part (a) to prove that any function can be written as a sum of even and odd functions. [Hint: Add the two equations in part (a).] (c) Use the result of part (b) to write each function as a sum of even and odd functions. \(f(x)=x^{2}-2 x+1, \quad k(x)=\frac{1}{x+1}\)

The winning times (in minutes) in the women's 400 -meter freestyle swimming event in the Olympics from 1948 through 2008 are given by the following ordered pairs. $$\begin{array}{lll} (1948,5.30) & (1972,4.32) & (1992,4.12) \\ (1952,5.20) & (1976,4.16) & (1996,4.12) \\ (1956,4.91) & (1980,4.15) & (2000,4.10) \\ (1960,4.84) & (1984,4.12) & (2004,4.09) \\ (1964,4.72) & (1988,4.06) & (2008,4.05) \\ (1968,4.53) & & \end{array}$$ A linear model that approximates the data is \(y=-0.020 t+5.00,-2 \leq t \leq 58,\) where \(y\) represents the winning time (in minutes) and \(t=-2\) represents 1948. Plot the actual data and the model on the same set of coordinate axes. How closely does the model represent the data? Does it appear that another type of model may be a better fit? Explain.

Use the given values of \(k\) and \(n\) to complete the table for the direct variation model \(y=k x^{n} .\) Plot the points in a rectangular coordinate system. $$\begin{array}{|l|l|l|l|l|l|}\hline x & 2 & 4 & 6 & 8 & 10 \\\\\hline y=k x^{n} & & & & & \\\\\hline\end{array}$$ $$k=\frac{1}{2}, n=3$$

Use Hooke's Law for springs, which states that the distance a spring is stretched (or compressed) varies directly as the force on the spring. The coiled spring of a toy supports the weight of a child. The spring is compressed a distance of 1.9 inches by the weight of a 25 -pound child. The toy will not work properly if its spring is compressed more than 3 inches. What is the maximum weight for which the toy will work properly?
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