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Chapter 3: Problem 13

Decide whether the given ordered pair is a solution of the given equation. $$ 2 x+y=5 ; \quad(3,-1) $$

Short Answer

Expert verified
(3, -1) is a solution to the equation.

Step by step solution

01

Understand the Equation and Ordered Pair

The given equation is a linear equation in two variables, which is represented as \(2x + y = 5\). The ordered pair provided is \((3, -1)\), where 3 is the value of \(x\) and -1 is the value of \(y\).
02

Substitute \(x\) and \(y\) into the Equation

To determine if \((3, -1)\) is a solution, substitute \(x = 3\) and \(y = -1\) into the equation \(2x + y = 5\). This gives: \[ 2(3) + (-1) \]
03

Simplify the Equation

Perform the arithmetic operation to simplify the equation after substitution: \[ 2(3) + (-1) = 6 - 1 = 5\]
04

Verify the Result

The simplified result is \( 5 \). Since this is equal to the right-hand side of the original equation \( 2x + y = 5 \), the ordered pair \((3, -1)\) satisfies 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 we talk about ordered pairs, we mean a pair of numbers used to locate a point on a coordinate plane. An ordered pair is written as \((x, y)\), where the first number represents the horizontal position (x-coordinate) and the second number represents the vertical position (y-coordinate).
In our exercise, the ordered pair given is \( (3, -1) \). This means we have to use 3 for x and -1 for y. By substituting these values into the given linear equation, we can check if the ordered pair is a solution. It's like testing if these coordinates fit perfectly into the equation's requirements.
substitution method
The substitution method is a straightforward way to determine if a pair of values works in an equation. Here鈥檚 how to approach it:
  • Take the given ordered pair and identify the values of x and y.
  • Replace (substitute) x and y in the given equation with these numbers.
  • Simplify the equation after substitution.
In the given problem, we substitute the values from the ordered pair \( (3, -1) \) into the equation 2x + y = 5. Replacing x with 3 and y with -1, we get:
\2(3) + (-1)\
Now, we simplify this to see if the left side equals the right side of the equation.
verifying solutions
Once you've substituted the x and y values into the equation, the next step is verifying if they make the equation true. This means simplifying the left-hand side and then comparing it to the right-hand side.
For our example, substituting x = 3 and y = -1 gives:
\[ 2(3) + (-1) = 6 - 1 = 5 \]
Since our result, 5, is equal to the right-hand side of the original equation (which is also 5), this verifies that the ordered pair \( (3, -1) \) is indeed a solution to the equation \( 2x + y = 5 \).
Verifying solutions helps ensure that the pair fits the equation perfectly, confirming correctness in pinpointing the exact coordinate values.

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