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Chapter 1: Problem 58

Find the slope and \(y\)-intercept of the line, and draw its graph. \(y=\frac{2}{3} x-2\)

Short Answer

Expert verified
Slope: \(\frac{2}{3}\), Y-intercept: \(-2\).

Step by step solution

01

Identify the Equation Format

The given equation is in the slope-intercept form, which is \(y = mx + b\). In this form, \(m\) is the slope and \(b\) is the \(y\)-intercept.
02

Determine the Slope

From the equation \(y = \frac{2}{3}x - 2\), identify \(m\), which is the slope. Compare it with the format \(y = mx + b\). Here, \(m = \frac{2}{3}\). Thus, the slope is \(\frac{2}{3}\).
03

Determine the Y-Intercept

From the equation \(y = \frac{2}{3}x - 2\), find \(b\), which is the \(y\)-intercept. Compare it with the format \(y = mx + b\). Here, \(b = -2\). Thus, the \(y\)-intercept is \(-2\), which corresponds to the point \((0, -2)\) on the graph.
04

Graph the Line

Begin by plotting the \(y\)-intercept, which is the point \((0, -2)\), on the graph. Starting from this point, use the slope \(\frac{2}{3}\), which means for every 3 units you move right (along the x-axis), move 2 units up (along the y-axis). Plot the next point at \((3, 0)\). Draw a straight line through these points to represent the equation.

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Key Concepts

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

Understanding Linear Equations
Linear equations are mathematical expressions that describe a straight line. They typically appear in the format: \[y = mx + b\]where:
  • y represents the dependent variable.
  • x is the independent variable.
  • m denotes the slope of the line.
  • b is the y-intercept.
The slope (m) tells us how steep the line is, showing the rate of change between y and x. The y-intercept (b) points where the line crosses the y-axis. Linear equations serve as a basic building block for understanding more complex algebraic constructs. They're essential for modeling relationships where changes in one variable proportionally affect another.
Graphing Lines from Linear Equations
Graphing lines involves translating the equation onto a visual plot. It's a way to see the relationship between variables clearly. Start by identifying the y-intercept鈥攖he point where the line crosses the y-axis. For the equation:\[y = \frac{2}{3}x - 2\]begin at the point (0, -2) on the graph. This spot is your anchor. Moving ahead, utilize the slope to set additional points. Here, slope = \(\frac{2}{3}\)This means:
  • For every 3 units you go to the right on the x-axis, go 2 units up on the y-axis.
Use these directions to find the next point. In this case, move from (0, -2) to (3, 0). Connect these points with a straight edge to complete the line. Remember, each line represents all possible solutions (x, y) that satisfy the equation.
Understanding Slope and Y-Intercept
The slope and y-intercept are two crucial elements that define a line's specific orientation and position. Let's focus on what each component does:
Slope:\This is a measurement of how slanted a line is. The value \(\frac{2}{3}\) reveals the line rises 2 units vertically for every 3 units it moves horizontally. It's a predictor of direction and steepness:
  • A positive slope means the line ascends from left to right.
  • A negative slope means it descends.
Y-Intercept:\This pinpoint \(-2\) shows where the line crosses the vertical y-axis. It signifies the starting value of y when x is zero:
  • Helps in quickly drawing the line on a graph.
  • Makes predicting y values straightforward when calculating equations manually.
Grasping these components supports not only in plotting but also enhances deeper understanding of algebraic relationships and changes over time or conditions.

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