Problem 17 - Algebra 1

Question

Solve the equation and describe each step you use. $$ 5(y-2)=-2(12-9 y)+y $$

Step-by-Step Solution

Answer

The solution to the equation is \(y = 1\)

Step 1: Distributive Property

Apply the distributive property, which means multiplying each term inside the parentheses by the factor on the outside. This gives: \( 5y - 10 = -24 + 18y + y \)

Step 2: Combine like terms

Combine the like terms on both sides of the equation to simplify. This results in: \( 5y - 10 = 19y -24 \)

Step 3: Rearrange terms

Shift the 'y'-terms to one side and the constant terms to the other side. This gives: \( 5y - 19y = -24 + 10 \)

Step 4: Further Simplification

Simplify both sides of the equation by combining like terms which results in: \( -14y = -14 \)

Step 5: Solve for 'y'

Now, isolate 'y' by dividing both sides of the equation by -14. This gives: \( y = 1 \)

Key Concepts

Distributive Property

In algebra, the distributive property helps to simplify expressions and solve equations by distributing multiplication over addition or subtraction within parentheses. For example, in the equation \(5(y-2)\), we apply the distributive property by multiplying 5 with each term inside the parentheses, resulting in \(5y - 10\). Similarly, on the other side, \(-2(12-9y)\) is expanded to \(-24 + 18y\) by multiplying -2 with both 12 and \(-9y\). This step is crucial in simplifying complex expressions and making them easier to work with in subsequent steps.

Combining Like Terms

Combining like terms is a fundamental process in simplifying algebraic expressions. It involves bringing together terms that have the same variable component. This is done to condense expressions or equations into a simpler form. For instance, in the equation \(5y - 10 = -24 + 18y + y\), we first rewrite it as \(5y - 10 = 19y - 24\) by combining \(18y + y\) into \(19y\). This simplification reduces clutter in equations and aids in better visualization, making the process of solving them far more efficient.

Isolating Variables

Isolating variables is a strategy used to solve equations by getting the unknown variable on one side of the equation. This entails rearranging the terms, often through moving terms across the equation, which requires changing their signs in the process. From the equation \(5y - 19y = -24 + 10\), we isolate the \(y\)-terms to one side resulting in \(-14y\), and the constants to the opposite side, ultimately achieving the equation \(-14y = -14\). The goal is to have the variable 'y' on one side to determine its specific value.

Algebraic Simplification

Algebraic simplification is essential to making an equation straightforward to solve. After combining like terms and isolating the variable, the next step is often simplifying both sides of the equation further if necessary. In our example, from \(-14y = -14\), we simplify by dividing each side by \(-14\) to isolate and solve for 'y'. This leads us to the solution \(y = 1\). Simplification is the final step in clearing away unwanted complexity, making it possible to find the value of the variable easily.