Problem 66 - Prealgebra

Question

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

Step-by-Step Solution

Answer

When \(x = \frac{5}{2}\), then \(y = 1\).

Step 1: Substitute the value of x into the equation

Given the equation \(2x + 5y = 10\) and \(x = \frac{5}{2}\), substitute \(x = \frac{5}{2}\) into the equation. This gives us: \(2\left(\frac{5}{2}\right) + 5y = 10\).

Step 2: Simplify the equation

Simplify the expression \(2\left(\frac{5}{2}\right)\) to get 5. The equation now becomes \(5 + 5y = 10\).

Step 3: Isolate the term with y

Subtract 5 from both sides of the equation to isolate the term with \(y\). This results in \(5y = 5\).

Step 4: Solve for y

Divide both sides of the equation by 5 to solve for \(y\). This gives \(y = 1\).

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra. It involves finding the value(s) of variable(s) that satisfy the equation. In our example, the equation is a linear equation: \(2x + 5y = 10\). Solving it requires rearranging it to find the unknown, which is \(y\). The key steps in solving equations include:

• Identifying known and unknown variables.
• Performing operations to isolate and solve for the unknown.
• Substituting back into the original equation to verify if the solution is correct.

By understanding these steps, you can solve simple and complex equations effectively.

Substitution Method

The substitution method is a useful strategy for solving equations that involve two or more variables. It involves replacing one variable with a given value or another expression to simplify the problem. In our specific exercise, we use the substitution method by inserting the value \(x = \frac{5}{2}\) into the equation \(2x + 5y = 10\). This enables us to focus solely on solving for \(y\). Here's how it's done:

• Take the given equation: \(2x + 5y = 10\).
• Substitute the value of \(x\) into the equation: \(2\left(\frac{5}{2}\right) + 5y = 10\).

By replacing \(x\) with its value, we turn the equation into one with a single variable, making it easier to solve. This method is especially helpful in systems of equations where solutions are not immediately obvious.

Simplifying Equations

Simplifying equations is about making them easier to work with by reducing complexity. In our example, after substituting \(x\), we get \(2\left(\frac{5}{2}\right) + 5y = 10\). Simplifying \(2\left(\frac{5}{2}\right)\) gives us 5. This makes the equation less cumbersome: \(5 + 5y = 10\).Simplification often involves:

• Performing arithmetic operations.
• Combining like terms.
• Reducing fractions to simpler forms.

By simplifying equations, you can strip away unnecessary complexity, which makes it easier to isolate and solve for variables.

Prealgebra

Prealgebra is the foundation for all higher-level mathematics, focusing on basic algebraic concepts. It covers understanding variables, basic equations, and arithmetic operations. In this exercise, we're working on a prealgebra concept by dealing with linear equations. Important prealgebra skills include:

• Understanding variables and coefficients.
• Recognizing operations needed to solve equations.
• Knowing how to substitute and simplify expressions.

Mastering prealgebra is crucial as it prepares students for more advanced topics in algebra and beyond. It equips learners with problem-solving skills that are applicable in everyday situations.