Problem 74 - Algebra 1

Question

Solve the equation. $$ 7 y=9 y-8 $$

Step-by-Step Solution

Answer

y = 4

Step 1: Simplify the equation

Combine like terms in the equation \(7y = 9y - 8\) by subtracting \(7y\) from both sides of the equation. This results in the equation \(0 = 2y - 8\).

Step 2: Solve the simplified equation

Solve the equation \(0 = 2y - 8\) for y by adding 8 to both sides to isolate \(2y\) on one side of the equation. This results in the equation \(8 = 2y\). Divide each side of this equation by 2 to solve for y, resulting in \(y = 4\).

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In algebra, letters and other symbols represent numbers and quantities in formulas and equations. This allows us to work with unknown values that we often aim to solve. By using algebra, we can create equations to describe relationships and use these equations to find specific values.

When dealing with equations, the primary goal is to isolate the variable (e.g., \(y\) in the given problem) on one side to find its value. For example, in the equation \(7y = 9y - 8\), the letter \(y\) is a variable representing a number that we want to find.

Algebra is powerful because it lets you solve real-world problems by translating them into mathematical equations, manipulating those equations, and then coming back to make a decision or prediction based on your findings.

Solving Equations

Solving equations is a fundamental skill in algebra. It involves finding the value of the unknown variable that makes the equation true. To solve an equation like \(7y = 9y - 8\), you need to perform steps that simplify the equation while retaining equality.

In our example, our first step is to get all terms involving \(y\) on one side of the equation. This means rearranging the equation so that like terms are together, making it easier to solve for \(y\). By subtracting \(7y\) from both sides, we simplify the equation to \(0 = 2y - 8\).

Next, we isolate the variable by performing operations to both sides. Adding 8 to both sides gives us \(8 = 2y\). The last step involves dividing each side by 2, resulting in \(y = 4\).

• Combine like terms.
• Use inverse operations to isolate the variable.
• Check your solution by substituting the variable back into the original equation.

By solving equations step-by-step, you ensure accuracy and a deeper understanding of the process involved.

Like Terms

Like terms are terms that contain the same variable raised to the same power. In simpler terms, they have the exact same variables and exponents, but they can have different coefficients (the numbers multiplying the variables). For example, in \(7y = 9y - 8\), the terms \(7y\) and \(9y\) are like terms because they both involve the variable \(y\).

Combining like terms is an essential step in simplifying equations. It helps reduce complex expressions into simpler forms, making solving the equation more straightforward.

When you combine like terms, you simply add or subtract the coefficients. For instance, if you have \(7y\) and \(9y\), subtracting \(7y\) from \(9y\) results in \(2y\).

• Identify terms with the same variable and power.
• Add or subtract their coefficients.
• Simplify the equation.

Understanding how to combine like terms is crucial in ensuring equations are simplified correctly and solutions are found easily.