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Chapter 3: Problem 23

Find the ordered pair solutions given the set of \(x\) -values. $$ y=-34 x+12 ;\\{-2,0,2\\} $$

Short Answer

Expert verified
The ordered pairs are \((-2, 80)\), \((0, 12)\), and \((2, -56)\).

Step by step solution

01

Plug in the first x-value

Start with the first value from the set, which is \( x = -2 \), and substitute it into the equation: \( y = -34(-2) + 12 \).
02

Calculate the first y-value

Simplify the expression to find the y-value: \( y = 68 + 12 = 80 \). So, the ordered pair with \( x = -2 \) is \((-2, 80)\).
03

Plug in the second x-value

Next, take the second value from the set, \( x = 0 \), and substitute it into the equation: \( y = -34(0) + 12 \).
04

Calculate the second y-value

Simplify to find the y-value: \( y = 0 + 12 = 12 \). So, the ordered pair for \( x = 0 \) is \((0, 12)\).
05

Plug in the third x-value

Finally, use the third value from the set, \( x = 2 \), and substitute it into the equation: \( y = -34(2) + 12 \).
06

Calculate the third y-value

Simplify the expression to find the y-value: \( y = -68 + 12 = -56 \). So, the ordered pair for \( x = 2 \) is \((2, -56)\).

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Key Concepts

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

Understanding Ordered Pairs
Ordered pairs are a fundamental part of graphing in mathematics. They are written in the form
  • \((x, y)\)
where the first value \(x\) represents the position along the horizontal axis and the second value \(y\) indicates the position along the vertical axis. Think of them as coordinates that tell you exactly where a point sits on a graph. For instance, the ordered pair
  • \((-2, 80)\) means that if you move left (since it's negative) to \(-2\) on the horizontal axis and then move up to \(80\) on the vertical axis, you'll find the point.
Each ordered pair corresponds to a unique point on this grid, allowing us to visually interpret relationships between variables.
Navigating the Coordinate System
The coordinate system is a grid that helps us locate points using ordered pairs. It consists of two number lines that intersect at a right angle:
  • The horizontal line, known as the x-axis.
  • The vertical line, known as the y-axis.
These lines divide the plane into four quadrants. Moving along the x-axis either gets you further left or right, while moving along the y-axis means you're traveling up or down. When plotting the ordered pairs from our exercise (
  • \((-2, 80)\)
  • \((0, 12)\)
  • \((2, -56)\)
), you'll see how they spread across these four quadrants, forming a visual representation of the equation \(y = -34x + 12\). This system makes it easier to understand how changes in \(x\) affect \(y\).
Exploring the Substitution Method
The substitution method is a technique we use to find corresponding ordered pairs for an equation. It involves replacing one variable with a given numerical value in order to solve for the other variable.
  • Begin by taking the given value of \(x\), substitute it into the equation, and solve for \(y\).
In our example, substituting each value from the set \(-2, 0, 2\) into \(y = -34x + 12\) allowed us to calculate different \(y\)-values.
  • For \(x = -2\), substitute to find \(y = 80\).
  • For \(x = 0\), substitute to find \(y = 12\).
  • For \(x = 2\), substitute to find \(y = -56\).
This method is highly effective for solving linear equations because it provides clear, step-by-step instructions for finding the solutions.

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