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Chapter 3: Problem 41

Find the slope and the \(y\) -intercept of the graph of \(x-3 y=7\) [ 2.3]

Short Answer

Expert verified
Slope is \(\frac{1}{3}\) and \(y\)-intercept is \(-\frac{7}{3}\).

Step by step solution

01

Rewrite in slope-intercept form

Start by rewriting the given equation in the slope-intercept form, which is \(y = mx + b\). The given equation is \(x - 3y = 7\). To isolate \(y\), first subtract \(x\) from both sides to obtain \(-3y = -x + 7\).
02

Solve for y

Now divide both sides by \(-3\) to solve for \(y\): \[y = \frac{-x + 7}{-3} \]Simplify the expression: \[y = \frac{x}{3} - \frac{7}{3} \].
03

Identify the slope and y-intercept

The equation is in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. Here, \(m = \frac{1}{3}\) and \(b = -\frac{7}{3}\).

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Key Concepts

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

Understanding 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. Linear equations take the form of sentences, and their solutions represent lines when plotted on a graph.
Common forms include the slope-intercept form (y = mx + b) and the standard form (Ax + By = C).
  • They have constant slopes.
  • The graph of a linear equation is a straight line.
Understanding linear equations is the foundation for working with slopes and intercepts.
Graphing Linear Equations
Graphing linear equations involves plotting points on the coordinate plane and drawing a line through these points. First, identify at least two points that satisfy the equation.
In the given problem, we convert the equation to slope-intercept form (y = mx + b), making it easier to determine key graphing elements:
  • The slope (m), which reflects the line's steepness and direction.
  • The y-intercept (b), where the line crosses the y-axis.
Using these components simplifies translating the equation into a visual graph.
Isolating Variables in Equations
Isolating variables means rearranging an equation to express one variable in terms of others. This process involves:
  • Using addition or subtraction to move terms from one side of the equation to another.
  • Employing multiplication or division to simplify the isolated variable.
In our example, we start with x - 3y = 7. By following these steps, we isolate y:
  • Subtract x from both sides: -3y = -x + 7.
  • Divide both sides by -3: y = (x/3) - (7/3).
This method is crucial for solving and understanding equations.
Solving for y in Slope-Intercept Form
The slope-intercept form, y = mx + b, provides clear information about the slope and y-intercept of a line. To solve for y in this form:
First, isolate y by moving other terms to the right side of the equation.
For the equation x - 3y = 7:
  • Subtract x from both sides to get: -3y = -x + 7.
  • Divide by -3 to simplify: y = (1/3)x - 7/3.
Now, the slope (m) is 1/3, and the y-intercept (b) is -7/3. This transformed equation makes graphing straightforward and highlights the key elements of the line's behavior.

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Most popular questions from this chapter

Determine whether each pair of lines is parallel, perpendicular, or neither. [ 2.4] $$ \begin{aligned} &2 x-5 y=6\\\ &2 x+5 y=1 \end{aligned} $$

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