Problem 31 - Prealgebra

Question

Indicate which of the given ordered pairs are solutions for each equation. $$y=2 x+3 \quad(0,3),(5,4),(2,0)$$

Step-by-Step Solution

Answer

Only the ordered pair (0,3) is a solution.

Step 1: Analyze the Equation

The equation provided is a linear equation in the form of \(y = 2x + 3\). This means for any given \(x\), you can find \(y\) by substituting the \(x\) value into the equation and computing \(y\).

Step 2: Check First Ordered Pair

Substitute \(x = 0\) from the ordered pair \((0,3)\) into the equation: \(y = 2(0) + 3 = 3\). Because \(y = 3\) matches the \(y\) value of the pair, \((0,3)\) is a solution for the equation.

Step 3: Check Second Ordered Pair

Substitute \(x = 5\) from the ordered pair \((5,4)\) into the equation: \(y = 2(5) + 3 = 13\). The calculated \(y\) value (13) does not match the \(y\) value of the pair (4), so \((5,4)\) is not a solution to the equation.

Step 4: Check Third Ordered Pair

Substitute \(x = 2\) from the ordered pair \((2,0)\) into the equation: \(y = 2(2) + 3 = 7\). The calculated \(y\) value (7) does not match the \(y\) value of the pair (0), so \((2,0)\) is not a solution to the equation.

Key Concepts

Ordered Pairs

In mathematics, ordered pairs are a fundamental concept used to describe a relationship between two variables, often denoted as \(x, y\). Think of it as a pair where the first value represents the 'x-coordinate,' and the second value represents the 'y-coordinate.' This is important because it shows precisely where a point lies on a coordinate plane. For example, the ordered pair \(0, 3\) means that when \(x = 0\), \(y = 3\). Ordered pairs are crucial in graphing and solving equations, as they help determine whether certain points satisfy a given mathematical equation or not.

• An ordered pair is written as \( (x, y) \).
• They are used extensively in linear equations to express points.
• The order of the values matters; \( (x, y) \) is different from \( (y, x) \).

Understanding ordered pairs helps you easily navigate through many algebraic problems. It's much like finding the location of a place using GPS coordinates. If you enter the right coordinates (or use the right ordered pair), you'll reach your destination (or find the right point that fits the equation).

Solution Checking

Solution checking involves verifying if certain values satisfy a given mathematical equation. It’s like a detective job. You take a possible solution and test it to see if it holds true. In the context of linear equations, you do this by substituting the candidate values into the equation and seeing if both sides balance.

• Take the given ordered pair, such as \( (0, 3) \).
• Substitute \(x\) and \(y\) values into the equation, e.g., \(y = 2x + 3\).
• Check if both left and right sides of the equation yield the same result.
• If they do, then the ordered pair is a solution to the equation.

For example, substitute \(x = 0\) and \(y = 3\) into the equation \(y = 2x + 3\), and you get \(3 = 2(0) + 3\), which simplifies to \(3 = 3\). This confirms the pair \( (0, 3) \) is a correct solution. Thus, solution checking assures you of the accuracy of your findings.

Substitution Method

The substitution method is a powerful tool in algebra that involves replacing a variable with a specific value in order to solve an equation. This technique helps us figure out if a given ordered pair is a solution to a linear equation. To carry out the substitution method, follow these simple steps:

• Identify an ordered pair that you want to test, such as \( (5, 4) \).
• Take the \(x\)-value from the ordered pair and substitute it in place of \(x\) in the equation \(y = 2x + 3\).
• Calculate the right side of the equation to find the value of \(y\).
• Compare the calculated \(y\) with the \(y\)-value from the ordered pair.

For example, replace \(x = 5\) in the equation \(y = 2(5) + 3\) to find \(y = 13\). If \(y = 13\) doesn't match the \(y\) of the ordered pair, \(y = 4\), then \( (5, 4) \) is not a solution. By using the substitution method, solving equations and verifying solutions becomes a structured and reliable process.