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Chapter 9: Problem 45

For Exercises \(41-60\), find the coordinates of the \(x\) - and \(y\) -intercepts. $$\frac{3}{4} x-y=-6$$

Short Answer

Expert verified
Answer: The x-intercept is (-8, 0) and the y-intercept is (0, 6).

Step by step solution

01

Find the x-intercept (when y=0)

To find the x-intercept, set y=0 in the given equation and solve for x: $$\frac{3}{4}x-(0)=-6$$ $$\frac{3}{4}x=-6$$
02

Solve for x

In order to solve for x, we can multiply both sides by the reciprocal of \(\frac{3}{4}\), which is \(\frac{4}{3}\): $$\frac{4}{3}\cdot\frac{3}{4}x=\frac{4}{3}\cdot(-6)$$ This simplifies to: $$x=-8$$
03

Write the x-intercept as a coordinate

The x-intercept is found when y=0, so the x-intercept coordinate is (-8, 0).
04

Find the y-intercept (when x=0)

To find the y-intercept, set x=0 in the equation and solve for y: $$\frac{3}{4}(0)-y=-6$$ $$-y=-6$$
05

Solve for y

To solve for y, divide both sides by -1: $$y=6$$
06

Write the y-intercept as a coordinate

The y-intercept is found when x=0, so the y-intercept coordinate is (0, 6). The x-intercept is (-8, 0) and the y-intercept is (0, 6).

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Key Concepts

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

x-intercept
Let's break down the concept of an x-intercept in the context of a linear equation. Simply put, the x-intercept is the point at which a graph crosses the x-axis. To find this point, we set the value of y to zero in the equation and solve for x.

For example, if we have the equation \( \frac{3}{4}x - y = -6 \), we would follow these steps:
  • Set y to 0, which gives us \( \frac{3}{4}x = -6 \).
  • Then, we solve for x by multiplying both sides by the reciprocal of \( \frac{3}{4} \) to get \( x = -8 \).
Therefore, the coordinates of the x-intercept for our equation are (-8, 0), indicating that the graph of our linear equation will cross the x-axis at the point \( x = -8 \).
y-intercept
The y-intercept is another critical point that is found on the line where it intersects the y-axis. To pinpoint the y-intercept, x is set to zero, and we solve for y in the linear equation.

Take our example equation \( \frac{3}{4}x - y = -6 \). By setting x to 0, we'll be left with \( -y = -6 \). To find y, simply transpose the negative sign by multiplying both sides by -1, yielding \( y = 6 \).
  • The coordinates for the y-intercept are (0, 6), showing where the line crosses the y-axis.
Understanding y-intercepts allows us to graph the vertical crossing point of linear equations easily.
solving linear equations
Solving linear equations is a foundational skill in algebra. These equations can always be rewritten in the standard form \( Ax + By = C \), where A, B, and C are constants. The solutions to these equations are the x and y values that make the equation true.

Here's a step-by-step approach when we're given \( \frac{3}{4}x - y = -6 \):
  • Pick a variable to solve for - in this case, we'll start with x.
  • Isolate x on one side by setting the other variable, y, to zero.
  • Perform algebraic operations to solve for x, which may include multiplying or dividing both sides by a constant.
  • Repeat similar steps to solve for y by setting x to zero.
Mastering these steps ensures that you can always find the x and y values that satisfy the linear equation.
graphing linear equations
Graphing linear equations involves plotting the relationship between two variables on a coordinate system. The graph is always a straight line. Every point on the line is a solution to the equation.

To graph our example \( \frac{3}{4}x - y = -6 \) equation, we'd:
  • Identify the x and y intercepts; we've found them to be (-8, 0) and (0, 6).
  • Plot these intercepts on a graph with a grid.
  • Draw a line through these two points, extending it to the edges of the grid.
The line crossing these intercepts represents all possible combinations of x and y that satisfy our equation. By understanding how to plot these graphs, students can better visualize and interpret the relationships between variables in linear equations.

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Most popular questions from this chapter

For Exercises \(1-12,\) determine whether the ordered pair is a solution for the equation. $$(-1.5,-1.3) ; y=0.2 x-1$$

For Exercises \(41-60\), find the coordinates of the \(x\) - and \(y\) -intercepts. $$5 x-4 y=20$$

For Exercises \(41-60\), find the coordinates of the \(x\) - and \(y\) -intercepts. $$x=-9$$

For Exercises \(1-12,\) determine whether the ordered pair is a solution for the equation. $$\left(\frac{2}{3},-4 \frac{5}{6}\right) ; y=\frac{1}{4} x-5$$

Find the midpoint of the given points. $$(1.5,-9) \text { and }(-4.5,2.2)$$
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