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Chapter 3: Problem 41

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ \frac{2}{3} x-3 y=7 $$

Short Answer

Expert verified
The x-intercept is \( \frac{21}{2} \) and the y-intercept is \( -\frac{7}{3} \).

Step by step solution

01

- Find the y-intercept

To find the y-intercept, set the value of x to 0 and solve for y.Start with the equation: \[\frac{2}{3} x - 3y = 7\]Substitute \( x = 0 \):\[\frac{2}{3}(0) - 3y = 7\]This simplifies to: \[-3y = 7\]Solving for y, we get: \[y = -\frac{7}{3}\]So, the y-intercept is \( (0, -\frac{7}{3}) \).
02

- Find the x-intercept

To find the x-intercept, set the value of y to 0 and solve for x.Starting with the equation: \[\frac{2}{3} x - 3y = 7\]Substitute \( y = 0 \):\[\frac{2}{3} x - 3(0) = 7\]This simplifies to: \[\frac{2}{3} x = 7\]Solving for x, we multiply both sides by \( \frac{3}{2} \): \[x=\frac{7 \cdot 3}{2}=\frac{21}{2}\]So, the x-intercept is \( (\frac{21}{2}, 0) \).
03

- Graph the equation

Now, plot the two intercepts on a coordinate plane.1. Plot the y-intercept at \( (0, -\frac{7}{3}) \).2. Plot the x-intercept at \( (\frac{21}{2}, 0) \).3. Draw a line through both points to represent the equation.This line represents all the solutions to the equation \( \frac{2}{3} x - 3 y = 7 \).

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Key Concepts

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

y-intercept
The y-intercept is the point where a graph crosses the y-axis. To find the y-intercept of an equation, set the value of x to 0 and solve for y. In the given equation \(\frac{2}{3} x - 3y = 7 \), we substitute x with 0, simplifying to \(-3y = 7\). Solving this equation by dividing both sides by -3 gives us y = \(-\frac{7}{3}\), meaning the y-intercept is at \((0, -\frac{7}{3})\).
x-intercept
The x-intercept is the point where a graph crosses the x-axis. To find the x-intercept, set y to 0 and solve for x. For the equation \(\frac{2}{3} x - 3 y = 7 \), we set y = 0. Simplifying the equation results in \(\frac{2}{3} x = 7\). By multiplying each side by \(\frac{3}{2}\), we solve for x, resulting in \(\frac{21}{2}\). Thus, the x-intercept is at \(\frac{21}{2}, 0\).
graphing linear equations
Graphing linear equations involves plotting points on a coordinate plane and connecting them with a straight line. For the equation \(\frac{2}{3} x - 3 y = 7 \), we first find the intercepts. We plot the y-intercept at \((0, -\frac{7}{3})\) and the x-intercept at \((\frac{21}{2}, 0)\). Connecting these points forms a straight line, representing all the solutions to the equation.
solving linear equations
Solving linear equations means finding values for variables that make the equation true. Usually, we find intercepts for graphing. For \(\frac{2}{3} x - 3 y = 7 \), solving for y when x = 0, and for x when y = 0, helps plot the necessary points for the graph. This process uses basics of algebra, such as substitution and solving for one variable.

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Most popular questions from this chapter

Segment PQ has the given coordinates for one endpoint P and for its midpoint M. Find the coordinates of the other endpoint \(Q .\) (Hint: Represent \(Q\) by \((x, y)\) and write two equations using the midpoint formula, one imolving \(x\) and the other involving \(y .\) Then solve for \(x\) and \(y .\) $$ P(2.5,1.75), M(3,2) $$

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. $$ \left(\frac{3}{4}, \frac{8}{3}\right) \text { and }\left(\frac{2}{5}, \frac{2}{3}\right) $$

Find an equation of the line that satisfies the given conditions. (a) Write the equation in slope-intercept form. (b) Write the equation in standard form. Through \((2,-7) ;\) perpendicular to \(5 x+2 y=18\)

Solve each problem. A package weighing \(x\) pounds costs \(f(x)\) dollars to mail to a given location, where f(x)=3.75 x $$ A. Evaluate \(f(3)\) B. Evaluate \(f(3)\) C. How much would it cost to mail a 5 -lb package? Interpret this question and its answer, using function notation. (PICTURES NOT COPY)

Solve each equation for \(y\). $$ 2 y-3 x=6 $$
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