Problem 69 - Prealgebra

Question

Use the formula \(2 x+5 y=10\) to find \(x\) if: $$y=-2$$

Step-by-Step Solution

Answer

The value of \(x\) is 10.

Step 1: Substitute the Given Value

We start with the equation \(2x + 5y = 10\). We know that \(y = -2\), so we substitute \(-2\) for \(y\) in the equation. This gives us \(2x + 5(-2) = 10\).

Step 2: Simplify the Equation

Simplify the equation by multiplying \(5\) and \(-2\): \(2x - 10 = 10\).

Step 3: Isolate the Variable Term

To isolate \(2x\), add \(10\) to both sides of the equation: \(2x - 10 + 10 = 10 + 10\), which simplifies to \(2x = 20\).

Step 4: Solve for x

Divide both sides by \(2\) to solve for \(x\): \(x = \frac{20}{2}\), which simplifies to \(x = 10\).

Key Concepts

Variable Isolation

In linear equations, variable isolation is an important technique. It helps you get the unknown variable on one side of the equation. This makes it easier to find its value. Consider the equation we are working with: \[2x + 5(-2) = 10.\]
For successful variable isolation, our goal is to get \(x\) by itself on one side. Here are the steps to follow:

• Look at the term with the variable you want to isolate. Here, it is \(2x\).
• Perform operations that simplify your equation and gradually move other terms to the opposite side.
• In this case, you add \(10\) to both sides to move the \(-10\) away: \[2x - 10 + 10 = 10 + 10.\]
• This operation results in \(2x = 20,\) with \(x\) now isolated for easy solving.

Equation Simplification

Equation simplification is about reducing equations to their simplest form while maintaining equality. A simple equation is easier to work with and understand. In our given problem, simplification appears in the early step.
We start with the substitute equation: \[2x + 5(-2) = 10.\]
The simplification process involves:

• Carefully multiply out constants with numbers that have been substituted. Here, multiply \(5\) by \(-2\) to simplify: \[2x - 10 = 10.\]
• This results in a less cluttered equation, setting up for effective isolation of the variable.
• At the end, solve the simplified equation by applying arithmetic operations appropriately.

Simplifying the equation makes it much easier to find the solution since it reduces the complexity of terms involved.

Substitution Method

The substitution method is a key strategy used to solve equations, especially when values for certain variables are already known. In our particular case, the given exercise provides that \(y = -2.\) This allows us to directly substitute this value into the original equation. The steps involved are:

• Begin with the original equation: \[2x + 5y = 10.\]
• Substitute \(-2\) for \(y\). This step changes the equation to: \[2x + 5(-2) = 10.\]
• With \(y\) substituted out, solve the equation as a one-variable equation, which simplifies the problem significantly.

The substitution method is effective because it leverages known values to simplify the problem, focusing attention directly on solving for the remaining unknowns.