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Chapter 3: Problem 37

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ x-3 y=6 $$

Short Answer

Expert verified
The x-intercept is \(6\) and the y-intercept is \(-2\).

Step by step solution

01

Find the \(x\)-intercept

The \(x\)-intercept occurs when \(y=0\). Substitute \(y=0\) into the equation \(x-3y=6\): \[x - 3(0) = 6\] Solve for \(x\): \[x = 6\] So, the \(x\)-intercept is \((6, 0)\).
02

Find the \(y\)-intercept

The \(y\)-intercept occurs when \(x=0\). Substitute \(x=0\) into the equation \(x-3y=6\): \[0 - 3y = 6\] Solve for \(y\): \[y = -2\] So, the \(y\)-intercept is \((0, -2)\).
03

Plot the Intercepts

Graph the equation by plotting the \(x\)-intercept \((6, 0)\) and the \(y\)-intercept \((0, -2)\) on the coordinate plane. Draw a straight line through these two points to represent the equation \(x - 3y = 6\).

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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 is the point where the graph of an equation crosses the x-axis. At this point, the value of y is zero. To find the x-intercept, we substitute y=0 into the equation and solve for x. For example, in the equation \(x - 3y = 6\), substituting \(y = 0\) gives us \(x - 3(0) = 6\). Simplifying it, we find that \(x = 6\). Hence, the x-intercept is \((6, 0)\). This point tells us where the graph touches the x-axis.
y-intercept
The y-intercept is where the graph crosses the y-axis. At this point, the value of x is zero. To find the y-intercept, we set x=0 in the equation and solve for y. For instance, in the equation \(x - 3y = 6\), substituting \(x = 0\) results in \(0 - 3y = 6\). Solving for y, we get \(y = -2\). So, the y-intercept is \((0, -2)\). This point shows where the graph meets the y-axis.
graphing linear equations
Graphing linear equations involves plotting points that satisfy the equation and connecting them to form a straight line. First, find the x- and y-intercepts. These points typically provide an easy way to start the graph. Next, plot these intercepts on a coordinate plane. For equation \(x - 3y = 6\), the x-intercept is \((6, 0)\) and the y-intercept is \((0, -2)\). Plot these points on the graph and draw a line through them. This line represents all the solutions to the equation.
finding intercepts
Finding intercepts is a crucial step in graphing equations. To locate the x-intercept, set y to zero and solve for x. For the y-intercept, set x to zero and solve for y. These intercepts help in sketching the basic structure of the graph. For example, for \(x - 3y = 6\), you get \(x = 6\) when \(y = 0\) (x-intercept) and \(y = -2\) when \(x = 0\) (y-intercept). These points are easier to calculate and plot accurately, giving a clear picture of the linear graph.

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Most popular questions from this chapter

Find the slope of the line through each pair of points.\(\left(\text {Hint:} \frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a}{b} \div \frac{c}{d}\right)\). $$ \left(\frac{3}{4}, \frac{1}{3}\right) \text { and }\left(\frac{5}{4}, \frac{10}{3}\right) $$

Find an equation of the line passing through the given points. (a) Write the equation in standard form. (b) Write the equation in slope-intercept form if possible. $$(6,1)\( and \)(-2,5)$$

An equation that defines \(y\) as a function fof \(x\) is given. (a) Solve for \(y\) in terms of \(x,\) and \(r e-\) place \(y\) with the function notation \(f(x) .\) (b) Find \(f(3) .\) See Example 6. $$ x-4 y=8 $$

Which of the following forms of the slope formula are correct? Explain. A. \(m=\frac{y_{1}-y_{2}}{x_{2}-x_{1}}\) B. \(m=\frac{y_{1}-y_{2}}{x_{1}-x_{2}}\) C. \(m=\frac{x_{2}-x_{1}}{y_{2}-y_{1}}\) D. \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

An equation that defines \(y\) as a function fof \(x\) is given. (a) Solve for \(y\) in terms of \(x,\) and \(r e-\) place \(y\) with the function notation \(f(x) .\) (b) Find \(f(3) .\) See Example 6. $$ x+3 y=12 $$
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