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Chapter 2: Problem 42

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ \frac{5}{7} x+\frac{6}{7} y=-2 $$

Short Answer

Expert verified
The x-intercept is \( -\frac{14}{5} \) and the y-intercept is \( -\frac{7}{3} \). Plot these points and draw the line through them.

Step by step solution

01

Identify X-intercept

To find the x-intercept, set y to 0. The x-intercept is the point where the line crosses the x-axis.
02

Solve for X-intercept

Substitute y = 0 into the equation: \[ \frac{5}{7}x + \frac{6}{7}(0) = -2 \] This simplifies to: \[ \frac{5}{7}x = -2 \] Now solve for x: \[ x = -2 \times \frac{7}{5} = -\frac{14}{5} \] So, the x-intercept is at \( x = -\frac{14}{5} \).
03

Identify Y-intercept

To find the y-intercept, set x to 0. The y-intercept is the point where the line crosses the y-axis.
04

Solve for Y-intercept

Substitute x = 0 into the equation: \[ \frac{5}{7}(0) + \frac{6}{7}y = -2 \] This simplifies to: \[ \frac{6}{7}y = -2 \] Now solve for y: \[ y = -2 \times \frac{7}{6} = -\frac{7}{3} \] So, the y-intercept is at \( y = -\frac{7}{3} \).
05

Plot the Intercepts

Plot the points \( (-\frac{14}{5}, 0) \) and \( (0, -\frac{7}{3}) \) on a coordinate plane.
06

Draw the Line

Draw a straight line through both points to represent the equation \( \frac{5}{7} x + \frac{6}{7} y = -2 \).

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Key Concepts

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

x-intercept
An x-intercept is the point where a graph crosses the x-axis. To find the x-intercept of an equation, we set the value of y to 0 and solve for x. This is because on the x-axis, the y-coordinate is always 0.

For the given equation \(\frac{5}{7}x + \frac{6}{7}y = -2\), we substitute y = 0 into the equation and simplify: \(\frac{5}{7}x + \frac{6}{7}(0) = -2\) becomes \(\frac{5}{7}x = -2\).

Next, we solve for x: \ x = -2 \times \frac{7}{5} = -\frac{14}{5}. Therefore, the x-intercept is \(x = -\frac{14}{5}\). This means the graph will cross the x-axis at the point \(-\frac{14}{5}, 0\).
y-intercept
The y-intercept is the point where a graph crosses the y-axis. To find the y-intercept of an equation, we set the value of x to 0 and solve for y. This is because on the y-axis, the x-coordinate is always 0.

For the equation \(\frac{5}{7}x + \frac{6}{7}y = -2\), we substitute x = 0 and simplify: \(\frac{5}{7}(0) + \frac{6}{7}y = -2\) simplifies to \(\frac{6}{7}y = -2\).

Solving for y, we get \ y = -2 \times \frac{7}{6} = -\frac{7}{3}. Therefore, the y-intercept is \( y = -\frac{7}{3}\). This means the graph crosses the y-axis at \(0, -\frac{7}{3}\).
solving linear equations
Solving linear equations involves finding the values of variables that satisfy the equation. A linear equation typically takes the form \ ax + by = c \. Here are the steps to solve for one specific variable:

  • Identify the variable to solve for
  • Isolate the variable by performing arithmetic operations (addition, subtraction, multiplication, division)
  • Ensure you perform the same operations on both sides
In our example, to find the x-intercept of \(\frac{5}{7}x + \frac{6}{7}y = -2\), we isolate x by setting y = 0 and solving for x. Similarly, to find the y-intercept, we set x = 0 and solve for y. The solutions to these equations give us the intercepts.
graphing equations
Graphing equations involves plotting points that satisfy the equation and then drawing a line through these points. Here鈥檚 how to graph a linear equation:

  • Find the x-intercept and y-intercept.
  • Plot these points on a coordinate plane.
  • Draw a straight line that passes through both intercepts.
For our given equation, we found the intercepts as \(-\frac{14}{5}, 0\) and \(0, -\frac{7}{3}\). By plotting these points on a coordinate plane and drawing a line through them, we visually represent the equation \( \frac{5}{7} x + \frac{6}{7} y = -2 \). This line shows all possible solutions for the equation.

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Most popular questions from this chapter

Graph each line passing through the given point and having the given slope. (0,0)\(; m=\frac{5}{3}\)

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