Problem 67 - Prealgebra

Question

Use the formula \(2 x+5 y=10\) to find \(x\) if: $$y=0$$

Step-by-Step Solution

Answer

If \( y = 0 \), then \( x = 5 \).

Step 1: Substitute the given value into the equation

We begin by substituting the value of \( y = 0 \) into the equation \( 2x + 5y = 10 \). This gives us \( 2x + 5(0) = 10 \).

Step 2: Simplify the equation

Simplify the equation from Step 1. This means we have \( 2x + 0 = 10 \), which simplifies to \( 2x = 10 \).

Step 3: Solve for \( x \)

To isolate \( x \), divide both sides of the equation \( 2x = 10 \) by 2. This results in \( x = \frac{10}{2} \), so \( x = 5 \).

Key Concepts

Step-by-step math solutions

Breaking down a math problem into clear, manageable steps can make solving equations much easier. Step-by-step solutions guide you through the process, one part at a time, helping to prevent mistakes and solidify your understanding.
To start, identify the problem and what is known and unknown. Take our example: the equation is \( 2x + 5y = 10 \) and we need to find \( x \) when \( y = 0 \).

Each step focuses on a single goal, such as substituting values or isolating a variable. With practice, these structured steps become second nature. Here's a tip: always check your work by substituting your solution back into the original equation to ensure it holds true.

Algebraic manipulation

Algebraic manipulation involves rearranging equations to either simplify them or to isolate a particular variable, using basic algebra skills. It's like solving a puzzle, where each move gets you closer to the solution.
Consider the equation \( 2x + 5y = 10 \). By substituting \( y = 0 \), you can simplify the expression. The equation becomes \( 2x + 0 = 10 \), which further simplifies to \( 2x = 10 \).
Here’s a quick checklist for algebraic manipulation:

• Combine like terms.
• Use inverse operations to move terms.
• Multiply or divide every part of an equation equally.

Algebraic manipulation is a powerful tool, not just for solving equations, but for simplifying mathematical expressions and solving real-world problems mathematically.

Substitution method

The substitution method is a technique used to solve systems of equations or expressions by replacing variables with given values. This method simplifies problems by reducing the number of unknowns.
In our example, \( y \) is given as 0. By substituting \( y = 0 \) into the equation \( 2x + 5y = 10 \), the equation then reduces to \( 2x = 10 \).
After substitution, you often find that solving for the remaining variables becomes straightforward. Here’s how it works:

• Identify the given variable value(s).
• Substitute these into the original equation.
• Simplify the equation to solve for the unknowns.

The substitution method helps break down complex equations, making them easier and quicker to solve, especially useful in finding specific solutions in algebraic problems.