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Chapter 3: Problem 7

Determine whether each equation has the given ordered pair as a solution. \(y=4 x-7 ;(2,1)\)

Short Answer

Expert verified
Yes, (2,1) is a solution.

Step by step solution

01

- Understand the equation and the ordered pair

The equation given is a linear equation in the form of \(y = 4x - 7\). The ordered pair given is (2,1), which means \(x = 2\) and \(y = 1\).
02

- Substitute the values of the ordered pair into the equation

Replace \(x\) with 2 and \(y\) with 1 in the equation \(y = 4x - 7\). This gives: \[1 = 4(2) - 7\]
03

- Simplify the equation

Calculate the right-hand side of the equation: \[4(2) = 8\] so the equation becomes \[1 = 8 - 7\]
04

- Verify the result

Simplify the right-hand side further: \[8 - 7 = 1\], so the equation becomes \[1 = 1\]. Since the left-hand side equals the right-hand side, the ordered pair (2,1) is a solution to the equation \(y = 4x - 7\).

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Key Concepts

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

Ordered Pairs
An ordered pair is simply a set of numbers used to locate a point on a coordinate plane. It's written in the form \((x, y)\), where \('x'\) is the x-coordinate (horizontal position) and \('y'\) is the y-coordinate (vertical position). For example, in the ordered pair (2, 1), the \'x\' value is 2 and the \'y\' value is 1.
Understanding ordered pairs is crucial in solving linear equations because it helps identify specific points we can use to test our equations. If the point satisfies the equation, it lies on the line represented by the equation.
Substitution
Substitution is a technique where you replace a variable in an equation with a given value. In our exercise, we use the ordered pair (2, 1) to substitute \(x = 2\) and \(y = 1\) into the equation. This method allows us to test whether the given pair is a solution.
For example, in the equation \y = 4x - 7\, we substitute \(x\) with 2 and \(y\) with 1:
\[ 1 = 4(2) - 7 \]
This substitution helps make the equation easier to evaluate and verify.
Simplification
Simplification involves performing the arithmetic operations to make an equation more manageable. In our problem, we substituted the ordered pair into the equation, transforming it into:
\[ 1 = 4(2) - 7 \] We then simplify it step-by-step:
  • First, calculate the multiplication: \[ 4(2) = 8 \]
  • Next, do the subtraction: \[ 8 - 7 = 1 \]
    • Now, the equation simplifies to \[ 1 = 1 \], which is easy to understand and shows the equality is valid.
Solution Verification
Solution verification is the process of ensuring that the values substituted into the equation produce a true statement. After substitution and simplification, we verify whether the left-hand side equals the right-hand side. If they are equal, the ordered pair is indeed a solution.
In our case, after simplifying, the equation became:
\[ 1 = 1 \] Since the left side equals the right side, the ordered pair (2, 1) is confirmed to be a solution of the equation \y = 4x - 7\.
This verification step is essential as it assures us that our calculations are correct and the ordered pair meets the requirements of the equation.

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