Problem 10 - Algebra 2

Question

Solve each equation. Check your solution. $$ 4(q-1)-3(q+2)=25 $$

Step-by-Step Solution

Answer

The solution is \( q = 35 \).

Step 1: Distribute the Constants

First, distribute the 4 and -3 across the terms inside the parentheses. Apply the distributive property:\[4(q-1) - 3(q+2) = 25\]This becomes:\[4q - 4 - 3q - 6 = 25\]

Step 2: Combine Like Terms

Now, combine the like terms on the left side of the equation, which are the terms involving \( q \) and the constant terms:\[(4q - 3q) + (-4 - 6) = 25\]This simplifies to:\[q - 10 = 25\]

Step 3: Solve for q

To isolate \( q \), add 10 to both sides of the equation:\[q - 10 + 10 = 25 + 10\]This gives:\[q = 35\]

Step 4: Check the Solution

Substitute \( q = 35 \) back into the original equation to verify:\[4(35-1) - 3(35+2)\]Simplify each part:\[4 \times 34 - 3 \times 37 = 136 - 111\]Calculate:\[136 - 111 = 25\]Since the original equation holds true, the solution \( q = 35 \) is correct.

Key Concepts

Distributive Property

The distributive property is a useful tool that allows us to simplify expressions where a term is multiplied by a sum or difference inside parentheses. In mathematical terms, when you have a form like \( a(b + c) \), it can be rewritten as \( ab + ac \). This rule helps us expand expressions so we can work with them more easily. For example, in the equation \( 4(q-1) - 3(q+2) = 25 \), we apply the distributive property as follows:

• Distribute the 4 into \( (q-1) \), resulting in \( 4q - 4 \).


• Distribute the -3 into \( (q+2) \), resulting in \( -3q - 6 \).

This step transforms the equation into \( 4q - 4 - 3q - 6 = 25 \). The distributive property is vital for breaking down expressions, allowing us to address each term individually in the equation.

Like Terms

Combining like terms is an essential skill in simplifying algebraic expressions. Like terms have the same variables raised to the same powers, but they might have different coefficients. For instance, in the expression \( 4q - 3q \), both terms involve the variable \( q \). The coefficients, 4 and -3, are different, but since they are like terms, they can be easily combined. This results in \( 4q - 3q = q \).

• Focus on variable parts: Combine the coefficients of \( q \) which gives us \( q \).


• Look at constant terms: Combine \(-4\) and \(-6\) to obtain \(-10\).

After combining all the like terms in our equation, we simplify it to \( q - 10 = 25 \). This simplification is crucial as it reduces the complexity of the equation, making the subsequent steps more straightforward.

Isolating Variables

To solve an equation for a variable, our main goal is to isolate that variable on one side of the equation. This means moving all other terms to the opposite side, usually through addition or subtraction. In our equation \( q - 10 = 25 \), we want to isolate \( q \), so we add 10 to both sides:

• Add 10 to \( q - 10 \) which results in \( q \).


• Add 10 to 25 on the other side which gives 35.

This gives us \( q = 35 \). Isolating the variable is essential as it helps to find the solution for \( q \). This process involves reversing operations like addition, subtraction, multiplication, or division that are affecting the variable you're solving for.

Verification of Solutions

After solving an equation, it’s important to verify that your solution is correct by substituting it back into the original equation. This ensures that our solution maintains the balance of the equation. For the equation \( 4(q-1) - 3(q+2) = 25 \), we substitute \( q = 35 \):

• Calculate \( 4(35-1) = 4 \times 34 \) which equals 136.


• Calculate \( -3(35+2) = -3 \times 37 \) which equals -111.

Add together 136 and -111, which gives 25, thus confirming that our solution is correct. Verification is the final step in problem-solving as it confirms the accuracy of our solution and increases confidence in our calculation.