Problem 80 - Algebra for College Students

Question

Complete the given ordered pairs so that each ordered pair satisfies the given equation. $$(-1,\quad),(\quad, 4), \quad y=\frac{1}{2} x+2$$

Step-by-Step Solution

Answer

The completed ordered pairs are \((-1, 1.5)\) and \((4, 4)\).

Step 1: Find the missing y-value for the pair (-1, )

Use the equation \[ y = \frac{1}{2}x + 2 \]. Substitute \(-1\) for x: \[ y = \frac{1}{2}(-1) + 2 \]. Calculate the value: \[ y = -\frac{1}{2} + 2 = 1.5 \]. Therefore, the complete ordered pair is (-1, 1.5).

Step 2: Find the missing x-value for the pair (, 4)

Use the equation \[ y = \frac{1}{2}x + 2 \]. Substitute \(4\) for y: \[ 4 = \frac{1}{2}x + 2 \]. Solve for x: Subtract 2 from both sides: \[ 2 = \frac{1}{2}x \], Multiply both sides by 2: \[ x = 4 \]. Therefore, the complete ordered pair is (4, 4).

Key Concepts

Understanding Ordered Pairs

Ordered pairs are fundamental in coordinate geometry. They are written as \( (x, y) \). The first number represents the position on the x-axis (horizontal), while the second number represents the position on the y-axis (vertical). Together, they show a specific point on a graph.
For example, in the ordered pair \( (-1, 1.5) \), -1 tells us how far to move left from the origin (0,0), and 1.5 tells us how far to move up. It's like giving directions to a spot on a map.

• Always check the order; x is always first, and y is second.
• Typically, ordered pairs are used to show solutions to equations.

Using the Substitution Method

The substitution method is a technique to find specific values by replacing variables with known numbers.
This is particularly useful for solving equations and completing ordered pairs.
Here's a detailed look at how it works:
We have the equation \( y = \frac{1}{2}x + 2 \). To find the missing y-value when \( x = -1 \), substitute -1 for x in the equation
\[ y = \frac{1}{2}(-1) + 2 \]
Calculate this step-by-step:
\[ y = -\frac{1}{2} + 2 \]
\[ y = 1.5 \]
Repeating this for different x or y values helps complete the pairs.

• Always use the equation provided.
• Replace the variable with the known value.
• Follow through with basic algebra to solve for the unknown.

Solving for Variables

When working with linear equations, you'll often need to solve for one variable. This means finding the value of that variable that makes the equation true.
Let's solve for the x-value when y is known: \(y = 4\) and the equation is \( y = \frac{1}{2}x + 2 \). Follow these steps:
1. Substitute \(4 \) for \( y \)
\[4 = \frac{1}{2}x + 2 \]
2. Isolate x by subtracting 2 from both sides: \[2 = \frac{1}{2}x \]
3. Solve for x by multiplying both sides by 2: \[ x = 4 \].

• Isolating variables often involves basic operations like addition, subtraction, multiplication, and division.
• It's crucial to perform the same operation on both sides of the equation to maintain equality.