Problem 6 - Elementary Algebra

Question

Solve and check each equation. $$ 4(y-5)-3 y=-1 \text { for } y. $$

Step-by-Step Solution

Answer

Answer: The value of y in the given equation is $$y = 19$$.

Step 1: Simplify the equation

Begin by applying the distributive property to the expression $$4(y-5)$$. $$ 4(y-5) - 3y = -1 $$ $$ 4y - 20 - 3y = -1 $$

Step 2: Combine like terms

Combine the terms with the variable y. $$ (4y-3y) - 20 = -1 $$ $$ y - 20 = -1 $$

Step 3: Isolate the variable y

Add 20 to both sides of the equation to isolate y. $$ y = -1 + 20 $$

Step 4: Solve for y

Add the numbers on the right side of the equation. $$ y = 19 $$

Step 5: Check the solution

Plug the solution back into the original equation to ensure that it works. $$ 4(19-5) - 3(19) = -1 $$ $$ 4(14) - 57 = -1 $$ $$ 56 - 57 = -1 $$ $$ -1 = -1 $$ Since the equation holds true, we can conclude that the solution for y is correct. The solution for the given equation is $$y = 19$$.

Key Concepts

Algebraic Expressions

Algebraic expressions are the cornerstone of algebra. They are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. In our example, 4(y-5)-3y is an algebraic expression that represents a relationship involving a variable, y. Here, we see terms combined by subtraction and multiplication, which work together to convey a more complex mathematical idea. When solving linear equations, understanding how to manipulate these expressions is crucial.

Think of each term in an algebraic expression as a separate piece or component. In our original equation, 4(y-5) and -3y are two components you work with. The goal is to simplify and transform these expressions to find the value of the variable, in this case, y.

Distributive Property

When we talk about the distributive property in algebra, we're referring to the way multiplication is distributed over addition or subtraction. It's the rule that tells us how to handle parenthesis when a number is outside of them. Given the expression a(b + c), the distributive property allows us to expand this to ab + ac.

This property is vital for simplifying complex expressions. In the exercise, we apply it first: 4(y - 5) becomes 4y - 20. By distributing the 4, we multiply it with each term inside the parenthesis, resulting in two terms that are easier to work with as we progress towards the solution.

Combining Like Terms

Combining like terms is an essential skill in algebra that simplifies expressions to make equations more manageable. Like terms have the same variable raised to the same power. For instance, 2y and 3y are like terms, but 2y and 2x are not.

In our exercise, after using the distributive property, we combine the like terms 4y and -3y to get y. This is because adding or subtracting coefficients is permissible when the variables they affect are the same. This process condenses the equation and moves us a step closer to isolating the variable—our ultimate goal.

Isolation of Variables

Isolating the variable, often the last major step in solving linear equations, involves getting the variable by itself on one side of the equation. The goal is to have the variable with a coefficient of 1, so you can clearly identify its value.

In this exercise, once we've combined like terms, we're left with y - 20 = -1. The next step is to add 20 to both sides. This moves all the numbers (constants) to one side and the variable to the other, resulting in y = 19. Now we have successfully isolated y, and we can easily read off the solution. Remember, each step in isolating the variable must preserve the equality, so whatever operation you perform on one side must also be done to the other.