Problem 68 - Prealgebra

Question

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

Step-by-Step Solution

Answer

When \( y = 2 \), \( x \) is 0.

Step 1: Substitute the Given Value of y

We start by substituting the given value of \( y \) into the equation. We know \( y = 2 \), so we replace \( y \) in the equation: \[ 2x + 5(2) = 10 \] This simplifies to: \[ 2x + 10 = 10 \]

Step 2: Simplify the Equation

Now simplify the equation by subtracting 10 from both sides. This will help us isolate the term with \( x \):\[ 2x + 10 - 10 = 10 - 10 \]Simplifying the left side gives:\[ 2x = 0 \]

Step 3: Solve for x

Now, solve for \( x \) by dividing both sides of the equation by 2:\[ \frac{2x}{2} = \frac{0}{2} \]This simplifies to:\[ x = 0 \]

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra. When solving an equation, our goal is to find the value of the variable that makes the equation true. In our example, the variable is \(x\).

Here's how it works step by step.

• First, we substitute any known values into the equation. This helps in simplifying the equation and makes our task easier.
• Next, we simplify the equation if possible. This could mean combining like terms or moving numbers to different sides of the equation.
• Finally, we solve for the variable. This often involves performing inverse operations like addition, subtraction, multiplication, or division to isolate the variable.

These steps ensure a clear path to finding the correct value for the variable, which keeps us organized and on track.

Prealgebra

Prealgebra is the foundation of all algebra studies, and it introduces the basic concepts students need to tackle more advanced topics later.

Prealgebra involves:

• Understanding numbers and how they work – integers, fractions, and decimals.
• Learning how to manipulate numbers with basic operations like addition, subtraction, multiplication, and division.
• Grasping the concept of variables and how they can stand in for unknown values in equations.

By mastering prealgebra, students gain the confidence to work with equations and expression setups, ensuring a smoother transition to more complex algebraic concepts. This foundation is essential for solving equations effectively.

Substitution Method

The substitution method is one of the key techniques used in algebra to solve equations, particularly when dealing with systems of equations. It involves replacing a variable with a value or another expression to simplify the problem.

Let’s break down the steps:

• First, solve one of the equations for one of the variables if it isn't already isolated.
• Then, substitute the expression (or value) into the other equation. This eliminates one variable, leaving an equation with a single variable.
• Solve this new equation for the remaining variable.
• Substitute back the found value to determine the other variable, if necessary.

This method is especially useful when one of the equations is already easy to solve for a variable. In our exercise, directly substituting the given value for \(y\) simplifies our work and leads us quickly to the solution, providing an efficient path to solve for \(x\).