Problem 51 - Algebra 1: Concepts and Skills

Question

Solve the equation. (Lesson 3.5) $$\frac{1}{4}(12 y-4)-2 y=-3(y-5)$$

Step-by-Step Solution

Answer

The solution to the equation is \(y = 4\).

Step 1: Distribute

First distribute \(\frac{1}{4}\) within the parentheses. This gives \(3y - 1 - 2y = -3(y-5)\)

Step 2: Combine Like Terms

Next, combine like terms on the left side of the equation to simplify. This gives \(y - 1 = -3(y-5)\)

Step 3: Expand Right Side

Expand the right side of the equation by distributing -3 within the parentheses. This gives \(y - 1 = -3y + 15\)

Step 4: Isolate y Terms

Bring like terms together to one side of the equation. Add 3y to both sides. This gives \(4y - 1 = 15\)

Step 5: Isolate y

Finally, isolate y by getting rid of -1 from the left side. Add 1 to both sides. This gives \(4y = 16\). Divide both sides by 4 to get \(y = 4\)

Key Concepts

Distributive Property

Understanding the distributive property is essential when solving linear equations; it allows you to simplify expressions that involve both multiplication and addition (or subtraction). The property states that for any real numbers, a, b, and c, the expression \(a(b + c)\) can be expanded to \(ab + ac\). For example, when you encounter an equation like \(\frac{1}{4}(12y - 4)\), you would distribute \(\frac{1}{4}\) to both 12y and -4. This simplifies to \(3y - 1\), making it easier to solve for y.

Always ensure that you apply the distributive property accurately to avoid errors in the subsequent steps. As seen in the exercise above, properly distributing values leads seamlessly to the next step, combining like terms.

Combining Like Terms

Often in algebra, you'll need to simplify an equation by combining like terms. Like terms are terms that have the same variables raised to the same power. In the process of solving \(\frac{1}{4}(12y - 4) - 2y \) for \(y\), you have the terms \(3y\) and \(2y\) on the same side of the equation. These terms are considered 'like terms' because they both contain the variable \(y\) to the first power. By combining \(3y\) and \(2y\), you can simplify the equation further, resulting in \(y - 1\).

Be careful to correctly combine coefficients and keep the variable parts unchanged. This is a pivotal step towards arriving at a more manageable form of the equation where the variable can be isolated with ease.

Isolate the Variable

The final goal in solving an equation is to isolate the variable, in this case, \(y\), to find its value. After simplifying the equation using the distributive property and combining like terms, you should work on getting \(y\) by itself. In our textbook example, we've reached the step where the equation is \(4y - 1 = 15\). To isolate \(y\), you need to perform operations that 'undo' anything attached to \(y\). This means you'll add or subtract terms from both sides to eliminate constants and divide or multiply to get rid of coefficients.

For instance, adding \(1\) to both sides of \(4y - 1 = 15\) and then dividing by \(4\) leads to the isolated variable \(y = 4\). It's crucial to perform the same operation on both sides to maintain the equality, and thus the correct answer. This final step gives you the solution to the equation and completes the process.