Problem 58 - Elementary Algebra

Question

For the following problems, solve the literal equations for the indicated variable. When directed, find the value of that variable for the given values of the other variables. Solve \(x=4 y+7\) for \(y\).

Step-by-Step Solution

Answer

Question: Solve for y in terms of x in the equation \(x = 4y + 7\). Answer: \(y = \frac{x-7}{4}\)

Step 1: Identify the given equation and the required variable

We are given the equation \(x=4y+7\). We need to solve it for y - that means, we want to get \(y\) alone on one side of the equation.

Step 2: Isolate the term containing the variable

We want y to be alone on one side of the equation. Looking at the equation \(x=4y+7\), we see that we need to get rid of the "+7" term. We can do so by subtracting 7 from both sides of the equation: \(x-7= 4y\)

Step 3: Solve for the variable

Now, we have the equation \(x-7=4y\). To solve for y, we'll need to divide both sides of the equation by 4: \(\frac{x-7}{4}=y\)

Step 4: Write the final expression for the variable

Now that we've isolated y, we can write our final expression for the variable y in terms of x: \(y=\frac{x-7}{4}\)

Key Concepts

Solving Equations

Solving equations is a fundamental skill in mathematics that involves finding the value of a variable that makes the equation true. Whether you are working with simple arithmetic equations or complex algebraic expressions, the goal is always to determine the value or values of the unknown variables.

In the original problem, the task was to solve for the variable \( y \). The equation given was \( x = 4y + 7 \). Solving an equation often involves performing operations to either simplify or transform the equation. For this equation, we follow a step-by-step approach, ensuring each action maintains the equality. This means whatever operation you perform on one side of the equation, you must perform on the other.

• Identify what is being asked and which variable you are solving for.
• Perform operations needed to isolate the variable.
• Always keep the equation balanced by doing the same operation to both sides.

Understanding the process of solving equations builds a strong foundation for more advanced algebraic problems.

Isolating Variables

Isolating variables is a crucial step in solving equations because it allows you to express one variable explicitly in terms of others. The goal in this process is to 'free' the variable from other terms that hold it, meaning to have the variable on one side of the equation all by itself.

In the example provided, we wanted to isolate \( y \) in the equation \( x = 4y + 7 \). To do this, we first needed to remove the constant term "7" that was added to the variable portion "4y". We achieved this by subtracting 7 from both sides, resulting in the equation \( x - 7 = 4y \).

• First, remove any constants on the same side as the variable.
• Next, handle coefficients that might be multiplying the variable.
• Use inverse operations, like subtraction to cancel addition, or division to cancel multiplication.

Isolating a variable allows you to re-write the equation in a format that is typically easier to interpret or manipulate for further calculations.

Algebraic Manipulation

Algebraic manipulation involves using various mathematical techniques to transform an equation or expression into a desired form. This allows for easier solving or further analysis of the problem. It is a skill that is used throughout mathematics, not only in algebraic contexts.

When manipulating the equation \( x = 4y + 7 \), the first step was subtracting 7 from both sides. This operation was aimed at isolating the term involving \( y \), resulting in \( x - 7 = 4y \). Next, we divided both sides by 4 to solve for \( y \), leading to the expression \( y = \frac{x - 7}{4} \).

• Use basic arithmetic operations (addition, subtraction, multiplication, division).
• Apply these operations to rearrange terms and simplifying fractions.
• Remember each manipulation must maintain the equation or expression's integrity.

Mastering algebraic manipulation is essential as it prepares students for more advanced topics and problem-solving scenarios in mathematics.