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Chapter 3: Problem 40

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ 3 x-7 y=9 $$

Short Answer

Expert verified
x-intercept: 3, y-intercept: -\frac{9}{7}.

Step by step solution

01

- Find the x-intercept

To find the x-intercept, set y to 0 and solve for x: \(3x - 7(0) = 9 \) \(3x = 9 \) \(x = \frac{9}{3} \) \(x = 3.\)
02

- Find the y-intercept

To find the y-intercept, set x to 0 and solve for y: \(3(0) - 7y = 9\) \(-7y = 9 \) \(y = \frac{9}{-7} \) \(y = -\frac{9}{7}.\)
03

- Plot the intercepts

Plot the x-intercept (3, 0) and the y-intercept (0, \(-\frac{9}{7}\)) on a coordinate plane.
04

- Draw the line

Draw a straight line passing through the points (3, 0) and (0, \(-\frac{9}{7}\)). This line represents the equation 3x - 7y = 9.

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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 linear equation is the point where the graph crosses the x-axis. This happens when the y-coordinate is zero. To find the x-intercept for the equation given in the exercise, follow these steps:

Set y to 0 in the equation:
$$3x - 7(0) = 9$$
Simplify the equation:
$$3x = 9$$
Solve for x by dividing both sides by 3:
$$x = \frac{9}{3}$$
$$x = 3$$

So, the x-intercept is the point (3, 0). This is where the graph will intersect the x-axis.
y-intercept
The y-intercept is the point where the graph crosses the y-axis. This happens when the x-coordinate is zero. To find the y-intercept for the equation given in the exercise, follow these steps:

Set x to 0 in the equation:
$$3(0) - 7y = 9$$
Simplify the equation:
$$-7y = 9$$
Solve for y by dividing both sides by -7:
$$y = \frac{9}{-7}$$
$$y = -\frac{9}{7}$$

So, the y-intercept is the point (0, -9/7). This is where the graph will intersect the y-axis.
coordinate plane
The coordinate plane is a two-dimensional surface used for graphing equations. It consists of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at the origin (0,0). Each point on the plane can be described by a pair of numbers, known as coordinates, which are written as (x, y). To plot points on a coordinate plane:

  • Find the x-coordinate and move along the x-axis to that value.
  • Find the y-coordinate and move along the y-axis to that value.
  • The intersection of these two values is the point's location.

In the exercise, we plot the x-intercept (3, 0) and the y-intercept (0, -9/7) on the coordinate plane, and draw a line through these points to graph the equation.
linear equations
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. They can be written in the form Ax + By = C, where A, B, and C are constants. Linear equations graph as straight lines, and their solutions are all the points on these lines.

In the exercise, we started with the linear equation 3x - 7y = 9. By finding the x- and y-intercepts, we identified two points that lie on the graph. Here's a recap:
  • Find where the line crosses the x-axis (x-intercept by setting y to 0).
  • Find where the line crosses the y-axis (y-intercept by setting x to 0).
  • Plot these intercepts on the coordinate plane and draw a line through them.

This process effectively graphs any linear equation, letting us visually represent solutions.

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Most popular questions from this chapter

For each situation, (a) write an equation in the form \(y=m x+b,(b)\) find and interpret the ordered pair associated with the equation for \(x=5,\) and \((c)\) answer the question. A ticket for the 2010 Troubadour Reunion, featuring James Taylor and Carole King. costs \(\$ 112.50 .\) A parking pass costs \(\$ 12 .\) (Source: Ticketmaster.) Let \(x\) represent the number of tickets and \(y\) represent the cost. How much does it cost for 2 tickets and a parking pass?

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 \((-2,-2) ;\) parallel to \(-x+2 y=10\)

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 involving \(x\) and the other involving \(y .\) Then solve for \(x\) and \(y .\) $$ P(5,8), M(8,2) $$

Write an equation in the form \(y=m x\) for each situation. Then give the three ordered pairs associated with the equation for x-values 0,5, and 10. See Example 7(a). \(x\) represents the number of t-shirts sold at \(\$ 26\) each, and \(y\) represents the total cost of the t-shirts (in dollars).

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ -\frac{9}{4} y=x $$
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