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Chapter 15: Problem 9

Determine whether each equation has the given ordered pair as a solution. \(y=2 x-5 ;(1,3)\)

Short Answer

Expert verified
(1, 3) is not a solution.

Step by step solution

01

Identify the Given Values

The given equation is \(y = 2x - 5\). The ordered pair to test is \((1, 3)\), where \(x = 1\) and \(y = 3\).
02

Substitute the Values

Substitute the values of \(x\) and \(y\) from the ordered pair into the equation. Replace \(x\) with \(1\) and \(y\) with \(3\). The equation becomes \(3 = 2(1) - 5\).
03

Simplify the Equation

Perform the arithmetic operations on the right side of the equation: \(3 = 2 \cdot 1 - 5\). Calculate \(2 \cdot 1\), which is \(2\). Then subtract \(5\) from \(2\) to get \(2 - 5 = -3\).
04

Compare Both Sides

After simplifying, we have \(3 = -3\). Since \(3\) does not equal \(-3\), the equation is not satisfied by the given ordered pair.
05

Conclusion

The ordered pair \((1, 3)\) is not a solution to the equation \(y = 2x - 5\).

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Key Concepts

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

Substitution Method
The substitution method is a key technique in solving linear equations. It's used to replace a variable with its given value. In the equation from the exercise, we use the ordered pair \( (x, y) = (1, 3) \). The process is simple:
  • Identify the given values of the variables.
  • Substitute these values into the equation.
  • Simplify and compare both sides.
This method helps check if a specific ordered pair is a solution to the equation. Here, we started by identifying \(x = 1\) and \(y = 3\). Replacing these in \(y = 2x - 5\) gave us \3 = 2(1) - 5\. Simplifying this, we noticed that \3\ does not equal \-3\. Hence, the pair \( (1, 3) \) is not a solution.
Ordered Pairs
Ordered pairs are a fundamental concept in algebra that represents a point on the coordinate plane. Each pair is written in the form \( (x, y) \).
  • The first element is the x-coordinate.
  • The second element is the y-coordinate.
In the given exercise, the ordered pair is \( (1, 3) \). This means:
  • \(x = 1\)
  • \(y = 3\)
To verify if an ordered pair is a solution to an equation, substitute the x and y values into the equation. If both sides of the equation are equal after simplification, then the pair is a solution. Otherwise, it's not. For \( (1, 3) \), substituting into \( y = 2x - 5 \) results in \3 = 2(1) - 5\, leading to a false statement.
Algebraic Simplification
Algebraic simplification involves reducing an expression to its simplest form. To do this, follow these steps:
  • Perform arithmetic operations like multiplication or division.
  • Combine like terms.
In our problem, simplification was a key step. Starting with \ y = 2x - 5 \ and substituting \ x = 1 \, we had \ 2(1) \:
  • Perform the multiplication: \2 \times 1 = 2\.
  • Subtract 5 from 2: \ 2 - 5 = -3 \.
The resulting equation was \3 = -3\, which is not true. Therefore, the pair \( (1, 3) \) does not satisfy the equation. Properly simplifying each part ensures accuracy in determining solutions.

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