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Magazine Chapter 4: Problem 34
In the following exercises, find three solutions to each linear equation. \(y=3 x-9\)
Short Answer
Expert verified
(0, -9), (1, -6), (2, -3)
Step by step solution
01
Identify the Form of the Equation
The equation is in slope-intercept form, which is given by: \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept.
02
Choose a Value for x
Select a value for \( x \). For example, let \( x = 0 \).
03
Calculate y for x = 0
Substitute \( x = 0 \) into the equation: \[ y = 3 \times 0 - 9 = -9 \]So, the first solution is \( (0, -9) \).
04
Choose Another Value for x
Select another value for \( x \). For example, let \( x = 1 \).
05
Calculate y for x = 1
Substitute \( x = 1 \) into the equation: \[ y = 3 \times 1 - 9 = -6 \]So, the second solution is \( (1, -6) \).
06
Choose a Third Value for x
Select a third value for \( x \). For example, let \( x = 2 \).
07
Calculate y for x = 2
Substitute \( x = 2 \) into the equation: \[ y = 3 \times 2 - 9 = -3 \]So, the third solution is \( (2, -3) \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope-Intercept Form
The slope-intercept form of a linear equation is one of the key concepts in algebra. It is a straightforward way to express a linear relationship between two variables, typically x and y. The general formula is expressed as: \[ y = mx + b \]Here:
- \( m \) represents the slope, which indicates the steepness and direction of the line.
- \( b \) represents the y-intercept, the point where the line crosses the y-axis.
Solving Equations
Solving equations means finding the values of the variables that make the equation true. When dealing with linear equations in slope-intercept form, such as \( y = 3x - 9 \), the goal is usually to find pairs \( (x, y) \) that satisfy the equation.Here's a step-by-step method to solve such equations:
- Choose a value for \( x \). This value can be any real number. For example, let's choose \( x = 0 \).
- Substitute this value into the equation to find the corresponding \( y \) value. For \( x = 0 \), substitute as follows: \( y = 3 \times 0 - 9 = -9 \).
- Therefore, one solution is the coordinate pair \( (0, -9) \).
Finding Solutions
Finding solutions to a linear equation in slope-intercept form involves selecting values for one variable and solving for the other. This gives you points that lie on the line represented by the equation. For the equation \( y = 3x - 9 \), here are the steps to find three different solutions:
- **Choose a value for \( x \)**: Let's start with \( x = 0 \).
- **Calculate \( y \) for the chosen \( x \)**: Substitute \( x = 0 \) into the equation: \( y = 3 \times 0 - 9 = -9 \). One solution is \( (0, -9) \).
- **Choose another value for \( x \)**: Let鈥檚 select \( x = 1 \).
- **Calculate \( y \) for the second \( x \)**: Substitute \( x = 1 \) into the equation: \( y = 3 \times 1 - 9 = -6 \). A second solution is \( (1, -6) \).
- **Choose a third value for \( x \)**: Finally, select \( x = 2 \).
- **Calculate \( y \) for the third \( x \)**: Substitute \( x = 2 \) into the equation: \( y = 3 \times 2 - 9 = -3 \). A third solution is \( (2, -3) \).
