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Chapter 3: Problem 97

Write each statement as an equation in two variables. Then graph each equation. The \(y\) -value is 5 more than three times the \(x\) -value.

Short Answer

Expert verified
The equation is \( y = 3x + 5 \), and it graphs as a line with y-intercept 5 and slope 3.

Step by step solution

01

Understand the Problem

The problem provides a relationship between two variables, typically referred to as \( x \) and \( y \). Specifically, the \( y \)-value is said to be 5 more than three times the \( x \)-value.
02

Translate the Statement into an Equation

To write this as an equation in two variables, translate the words into a mathematical expression. "Three times the \( x \)-value" is written as \( 3x \). "5 more than" this product indicates addition, resulting in the equation:\[ y = 3x + 5 \]
03

Graphing the Equation

To graph the equation \( y = 3x + 5 \), start by identifying the y-intercept and slope. The equation is in slope-intercept form \( y = mx + b \), where \( m = 3 \) and \( b = 5 \). This means the line crosses the y-axis at \( (0, 5) \) and the slope of the line is 3. This signifies that for every unit increase in \( x \), \( y \) increases by 3.
04

Plot the Y-Intercept

Locate the point \( (0, 5) \) on a graph. This point is where the graph of the equation crosses the y-axis.
05

Use the Slope to Plot a Second Point

From the y-intercept \( (0, 5) \), apply the slope of 3 \( \left( \frac{3}{1} \right) \). Rise 3 units up and 1 unit right to locate the next point, \( (1, 8) \).
06

Draw the Line

Using the two points, \( (0, 5) \) and \( (1, 8) \), draw a straight line extending in both directions, which represents all the solutions of the equation \( y = 3x + 5 \).

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Key Concepts

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

Graphing Equations
Graphing equations is an essential skill in algebra. It helps you visualize relationships between variables. When you graph an equation, you're placing every possible solution on a coordinate plane.
For the equation in our original problem, the steps to graph begin with identifying crucial components: the y-intercept and the slope. These details tell you where the line starts and how it moves along the graph.
  • Y-intercept: This is the point where the line intersects the y-axis. For our equation, this is at the point (0, 5).
  • Slope: The slope indicates the steepness of the line and the direction it goes. A slope of 3 means the line goes up 3 units for every unit it moves to the right.
Start by plotting the y-intercept on the graph. Then use the slope to find the next points.
These points help sketch the line, revealing every solution to the equation. Visualizing this can make understanding relationships between variables more intuitive.
Slope-Intercept Form
The slope-intercept form is a way to express linear equations. It's written as: \[ y = mx + b \]Here, \( m \) represents the slope and \( b \) represents the y-intercept.This form is popular due to its simplicity. It directly shows how the line behaves:
  • The slope, \( m \), shows the relationship between the changes in \( x \) and \( y \).
  • The y-intercept, \( b \), shows where the line crosses the y-axis.
For example, in the equation \( y = 3x + 5 \), the slope \( m \) is 3, telling us for every increase in \( x \, \), \( y \) increases by 3. The y-intercept \( b \) is 5, showing us where the line hits the y-axis.
Recognizing this form in equations is key for quickly sketching lines and understanding linear relationships.
Two-Variable Equations
Two-variable equations illustrate how two quantities relate to each other. They often use variables like \( x \) and \( y \) to demonstrate this.
  • Variables: In our exercise, \( x \) and \( y \) represent different quantities whose relationship is defined by the equation.
  • Equation: Example: \( y = 3x + 5 \). This tells us how \( y \) depends on \( x . \)
These equations allow us to explore how modifying one variable impacts the other. They're fundamental to algebra and applied in various disciplines. By representing them graphically, you make it easier to see trends and predict outcomes.
Understanding two-variable equations helps solve real-world problems, where you often have two changing quantities influencing each other.

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