Problem 35 - Prealgebra

Question

Indicate which of the given ordered pairs are solutions for each equation. $$y=5 x \quad(0,5),(1,5),\left(2, \frac{5}{2}\right)$$

Step-by-Step Solution

Answer

The ordered pair \((1, 5)\) is a solution to the equation.

Step 1: Understand the Equation

The given equation is a linear equation in the form of \(y = 5x\). This means for every x-value, the y-value should be five times the x-value.

Step 2: Check First Ordered Pair

For the ordered pair \((0, 5)\), substitute \(x = 0\) into the equation: \(y = 5 \times 0 = 0\). This result is not equal to the y-value given, which is 5.

Step 3: Check Second Ordered Pair

For the ordered pair \((1, 5)\), substitute \(x = 1\) into the equation: \(y = 5 \times 1 = 5\). This result matches the given y-value, which is 5.

Step 4: Check Third Ordered Pair

For the ordered pair \((2, \frac{5}{2})\), substitute \(x = 2\) into the equation: \(y = 5 \times 2 = 10\). This result does not match the given y-value, which is \(\frac{5}{2}\).

Key Concepts

Understanding Ordered Pairs

Ordered pairs are an essential concept in mathematics, particularly when working with coordinate systems and equations. They are written in the form

• \((x, y)\)

where the first number represents the x-coordinate and the second number represents the y-coordinate. In the context of solving equations, ordered pairs are used to express solutions that satisfy a given equation.

For example, if you have a linear equation like \(y = 5x\), any ordered pair that satisfies this equation will be a solution. This means when you substitute the x-value into the equation, the resulting y-value should match the second number in the ordered pair.

The Process of Solving Equations

Solving equations involves finding values for the variables that make the equation true. In the given example of the linear equation, \(y = 5x\), we are checking which ordered pairs solve the equation. The process is simple:

• Substitute the x-value from the ordered pair into the equation.
• Multiply it by 5 (as per the given equation) to find the corresponding y-value.
• Check if this calculated y-value equals the y-value in the ordered pair.

If the y-values match, the ordered pair is a solution to the equation. In our example, when we substituted - \(x = 1\), we found \(y = 5\), matching the ordered pair \((1, 5)\), indicating it's a solution.

However, the other pairs, \((0, 5)\) and \((2, \frac{5}{2})\), did not satisfy the equation, meaning they are not solutions.

Multiplication in Linear Equations

Multiplication plays a crucial role in understanding linear equations, especially when the equation is in the form \(y = mx\), where \(m\) is the slope and represents a constant rate of change.

In our example equation \(y = 5x\), the number \(5\) is the slope, and it tells us how much y increases for every unit increase in x. This is often referred to as the 'rise over run'. Each time we increase x by one unit, y increases by 5 units. This predictable relationship is the hallmark of linear equations.

When solving such equations, it's important to correctly perform multiplication. For ordered pairs to be solutions of this equation, after substituting x, multiplying by 5 must give us the correct y-value as per the ordered pair.

Understanding this concept helps students not only solve specific equations but also grasp the underlying principles of how variables interact within linear relationships.