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Chapter 2: Problem 49

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ -\frac{2}{3} y=x $$

Short Answer

Expert verified
The x- and y-intercepts are both (0, 0). The graph crosses at the origin with a slope of -2/3.

Step by step solution

01

Find the y-intercept

To find the y-intercept, set x to 0 and solve for y. The equation is given as \(-\frac{2}{3} y = x\), so we substitute x = 0:\ \[-\frac{2}{3} y = 0\]. Solving for y gives y = 0. Therefore, the y-intercept is (0, 0).
02

Find the x-intercept

To find the x-intercept, set y to 0 and solve for x. Using the equation \(-\frac{2}{3} y = x\), we substitute y = 0: \ \[-\frac{2}{3} \times 0 = x\]. Solving for x gives x = 0. Therefore, the x-intercept is (0, 0).
03

Graph the equation

To graph the equation, plot the intercepts found in Steps 1 and 2. The x-intercept and y-intercept are both at the point (0, 0). Since both intercepts are the same, this point is where the line crosses both the x-axis and y-axis. The equation \(-\frac{2}{3}y = x\) represents a straight line through the origin with a slope of \(-\frac{2}{3}\). This means the line will go down 2 units for every 3 units it goes to the right. Draw a straight line through the origin with this slope.

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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 a line is the point where the line crosses the x-axis. To find this point, we set the value of y to 0 in the equation and solve for x. In our equation \(-\frac{2}{3} y = x\), if we set y to 0, it becomes \(-\frac{2}{3} \times 0 = x\), which simplifies to 0. Hence, the x-intercept for our equation is (0, 0).This point is where the line touches the x-axis. Remember, intersections with the axes often give us useful points for graphing.
y-intercept
The y-intercept of a line is the point where the line crosses the y-axis. To determine this point, we set the value of x to 0 in the equation and solve for y. Using our equation, \(-\frac{2}{3} y = x\), substituting x= 0 we get \(-\frac{2}{3} y = 0\). Solving for y gives us y = 0. Therefore, the y-intercept for our equation is also at (0, 0). This y-intercept is the point where the line touches the y-axis. Both the x- and y-intercepts are essential in providing foundational points for graphing the line.
slope
The slope of a line dictates how steep the line is and in which direction it tilts. It measures the rate of change of y with respect to x. In our equation \(-\frac{2}{3} y = x\), we can rewrite it to identify the slope, which is just the coefficient of the transformed term. By isolating y, we get y = \(-\frac{3}{2} x\). Here, the coefficient of x is -\frac{3}{2}, and this tells us the slope of the line. The slope of -\frac{3}{2} means that for every 3 units we move to the right, the line drops 2 units downwards. Knowing the slope helps us draw the line accurately after plotting the intercepts.

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Most popular questions from this chapter

Forensic scientists use the lengths of certain bones to calculate the height of a person. Two such bones are the tibia \((t),\) the bone from the ankle to the knee, and the femur \((r),\) the bone from the knee to the hip socket. A person's height \((h)\) in centimeters is determined from the lengths of these bones using the following functions. For men: \(\quad h(r)=69.09+2.24 r\) or \(\quad h(t)=81.69+2.39 t\) For women: \(\quad h(r)=61.41+2.32 r\) or \(h(t)=72.57+2.53 t\) (a) Find the height of a man with a femur measuring \(56 \mathrm{~cm}\). (b) Find the height of a man with a tibia measuring \(40 \mathrm{~cm} .\) (c) Find the height of a woman with a femur measuring \(50 \mathrm{~cm}\). (d) Find the height of a woman with a tibia measuring \(36 \mathrm{~cm}\).

Graph each line passing through the given point and having the given slope. (2,-5)\(; m=0\)

Graph the intersection of each pair of inequalities. $$ x+y \leq 1 \quad \text { and } \quad x \geq 1 $$

The average price of a movie ticket in 2004 was \(\$ 6.21 .\) In \(2016,\) the average price was \(\$ 8.65 .\) Find and interpret the average rate of change in the price of a movie ticket per year to the nearest cent.

The table represents a linear function. (a) What is \(f(2)\) ? (b) If \(f(x)=2.1,\) what is the value of \(x ?\) (c) What is the slope of the line? (d) What is the \(y\) -intercept of the line? (e) Using the answers from parts (c) and (d), write an equation for \(f(x)\). $$ \begin{array}{|c|c|} \hline x & y=f(x) \\ \hline-1 & -3.9 \\ \hline 0 & -2.4 \\ \hline 1 & -0.9 \\ \hline 2 & 0.6 \\ \hline 3 & 2.1 \\ \hline \end{array} $$
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