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Chapter 3: Problem 77

Explain why the \(x\)-intercept, the \(y\)-intercept, and one other point are not three different solutions of \(y=2 x\).

Short Answer

Expert verified
The x-intercept and y-intercept for the equation \(y = 2x\) are both (0,0), hence there are only two different points, not three.

Step by step solution

01

- Understanding Intercepts

The x-intercept is the point where the graph crosses the x-axis. At this point, y=0. Similarly, the y-intercept is the point where the graph crosses the y-axis, so x=0 there.
02

- Finding the Intercepts

For the x-intercept, set y=0 in the equation. So, \(0 = 2x\), which gives \(x = 0\). Hence, the x-intercept is (0,0). For the y-intercept, set x=0, which gives \(y = 2(0) = 0\). Thus, the y-intercept is also (0,0).
03

- Identifying One Other Point

Choose any other value of x to find another point. For example, if \(x = 1\), then \(y = 2(1) = 2\). Thus, another point is (1,2).
04

- Analyzing the Results

Since both intercepts are the same point (0,0), there are only two different points found: (0,0) and (1,2). Therefore, the x-intercept and the y-intercept are not three different solutions.

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Key Concepts

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

Understanding the x-intercept
The x-intercept is the point on a graph where the line crosses the x-axis. At this specific point, the y-value is always zero. To find the x-intercept for any linear equation, you simply set y equal to zero and solve for x.
For the given linear equation, \(y = 2x\), if we set \(y = 0\) and solve for x, we get:
\[0 = 2x\]
Solving for x gives:
\[x = 0\]
Thus, the x-intercept for the equation \(y = 2x\) is the point (0,0).
Remember, the x-intercept helps us understand where a line crosses the x-axis, which is an essential step in graphing and understanding the equation's behavior.
Understanding the y-intercept
The y-intercept is the point on a graph where the line crosses the y-axis. At this point, the x-value is always zero. Finding the y-intercept involves setting x equal to zero in the equation and solving for y.
For the linear equation \(y = 2x\), by setting \(x = 0\), we get:
\[y = 2(0) = 0\]
This means the y-intercept for the equation \(y = 2x\) is also the point (0,0).
The y-intercept gives vital information about the line's behavior, particularly its starting point when plotted on a graph. For many equations, the y-intercept differs from the x-intercept, but in this specific case, they are the same.
Finding Points on a Graph
To graph a linear equation such as \(y = 2x\), knowing the intercepts is helpful, but it's also essential to identify other points on the line.
Once you determine the intercepts, you can choose any x-value to find corresponding y-values. For instance, if we select an x-value of 1, we substitute it into the equation to find the y-value:
\[y = 2(1) = 2\]
This gives us the point (1,2), which we can plot along with the intercepts.
Finding multiple points ensures that we can accurately draw the line represented by the equation. This method verifies the line's slope and overall direction. Try different x-values to get several points, plot them, and connect the dots to form the line.

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Most popular questions from this chapter

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((-1,-3) ;(-4,-1)\)

The completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Find the \(x\)-intercept of \(9 x+2 y=36\). Incorrect Answer: \(9 x+2 y=36\) $$ \begin{aligned} 9(0)+2 y &=36 \\ 2 y &=36 \\ \frac{2 y}{2} &=\frac{36}{2} \\ y &=18 \end{aligned} $$ The \(x\)-intercept is \((0,18)\).

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(a) find three solutions of the equation. (b) graph the equation. \(y=-\frac{1}{3} x+5\)

Balanced Rock in Arches National Park is \(55 \mathrm{ft}\) tall and weighs 3500 tons. Find its height in meters. Round to the nearest tenth. \((1 \mathrm{~m} \approx 3.2808 \mathrm{ft}\).) (Source: www.desertusa .com)
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