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Chapter 3: Problem 27

Determine whether each equation has the given ordered pair as a solution. \(y=-\frac{3}{4} x+8 ;(-8,12)\)

Short Answer

Expert verified
No, the ordered pair \((-8,12)\) is not a solution to the equation.

Step by step solution

01

Understand the Problem

We need to determine if the ordered pair \((-8, 12)\) is a solution to the equation \(y=-\frac{3}{4}x+8\). This involves substituting the \(x\) and \(y\) values from the ordered pair into the equation and checking if the equation holds true.
02

Substitute the x-value

Substitute \(x = -8\) into the equation. This gives us: \[y = -\frac{3}{4}(-8) + 8\] Calculate \(-\frac{3}{4}(-8)\) to simplify the equation.
03

Calculate the Multiplication

Calculate \(-\frac{3}{4}(-8)\), which simplifies to \(6\) because: 1. \(-\frac{3}{4}\) multiplied by \(-8\) is \(\frac{3}{4} \times 8\).2. This equals \(6\) because \(3 \times 8 = 24\), and \(24 \div 4 = 6\).Thus, we have \(y = 6 + 8\).
04

Finalize the Calculation

Now, add \(6 + 8\) to see if it equals the \(y\)-value from the ordered pair: \[y = 6 + 8 = 14\] This gives us \(y = 14\), which we will compare to the \(y\)-value from the ordered pair.
05

Compare to Original y-value

The \(y\) we calculated from substituting \(x = -8\) into the equation is \(14\). The given \(y\)-value from the ordered pair is \(12\). Since \(12 eq 14\), the ordered pair \((-8,12)\) does not satisfy the equation.

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Key Concepts

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

ordered pairs
When dealing with equations, ordered pairs are a way to represent a solution. An ordered pair consists of two elements written in a specific order, often as \(x, y\).
These elements represent the coordinates on a graph, where \(x\) gives the position on the horizontal axis and \(y\) gives the position on the vertical axis.
  • The first number in the pair corresponds to the \(x\)-value.
  • The second number in the pair corresponds to the \(y\)-value.
Ordered pairs are crucial for verifying solutions to equations. In our exercise, we determined if the ordered pair \((-8, 12)\) is a solution by substituting these values into the equation. If both sides of the equation are equal, then the ordered pair is a solution.
linear equations
Linear equations are mathematical statements that show a relationship between two variables, usually \(x\) and \(y\). They form a straight line when graphed on a coordinate plane.
The general form of a linear equation can be written as \(y = mx + b\), where:
  • \(m\) is the slope of the line, determining how sharp it inclines or declines.
  • \(b\) is the y-intercept, the point where the line crosses the \(y\)-axis.
In the exercise equation \(y = -\frac{3}{4}x + 8\), the slope is \(-\frac{3}{4}\), indicating a downward slope.
The y-intercept is \(+8\), showing that the line crosses the \(y\)-axis at 8. Understanding these components helps in predicting how changes in \(x\) affect \(y\) and the overall line's behavior.
substitution method
The substitution method is a mathematical technique used to determine if an ordered pair is a solution to an equation. This involves taking the values from the ordered pair and replacing the corresponding variables in the equation with these values.
Here's a step-by-step on how it's applied:
  • Identify the \(x\) and \(y\) values in the ordered pair.
  • Substitute these values in place of \(x\) and \(y\) in the equation.
  • Simplify the equation to check if both sides are equal.
In the given exercise, substituting \(x = -8\) and \(y = 12\) into the equation resulted in unequal values, proving that the ordered pair was not a solution.
This systematic approach is essential for verifying solutions, especially when analyzing whether a point lies on the graphed line of an equation.

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