Problem 46 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ -7+8(5-3 q)=3(7-9 q) $$

Step-by-Step Solution

Answer

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

Step 1: Apply the distributive property

Apply the distributive property on both sides of the equation to simplify it. And it becomes: \(-7+40-24q=21-27q\).

Step 2: Combine like terms

Combine like terms on both sides. For the left hand side, combine -7 and 40 to get 33. So, it becomes: \(33-24q=21-27q\).

Step 3: Solve for q

To isolate q, add 27q to both sides, then we get: \(33+3q=21\). Next, subtract 33 from both sides to isolate q which will be: \(3q=-12\). Finally, divide both sides by 3 to solve for q and the solution will be: \(q=-4\)

Key Concepts

Distributive Property

The distributive property is an essential tool for simplifying expressions and solving equations. It allows you to multiply a single term by each term within a set of parentheses. In simpler terms, you "distribute" the number outside the parentheses to each number inside.
For an equation like \[-7+8(5-3q)=3(7-9q)\], you'll need to apply the distributive property to both sides:

• On the left, multiply 8 by both 5 and \(-3q\). This gives you \(8 \times 5 = 40\) and \(8 \times (-3q) = -24q\).
• On the right, multiply 3 by both 7 and \(-9q\). This results in \(3 \times 7 = 21\) and \(3 \times (-9q) = -27q\).

Thus, the distributed equation becomes:\[-7 + 40 - 24q = 21 - 27q\]
Mastering this property is crucial because it simplifies complex expressions, making it easier to advance to further steps, like combining like terms.

Combining Like Terms

When solving equations, it's important to combine like terms to simplify the problem. Like terms have the same variables raised to the same power, for instance, \(-7\) and \(40\) are like terms because they are constant numbers (no variables involved).
In the current equation \(-7 + 40 - 24q = 21 - 27q\), we only have to focus on the left side first. Add \(-7\) and \(40\) to get \(33\). This process is crucial in bringing the equation to its simplest form.
The equation now looks like:\[33 - 24q = 21 - 27q\]
By combining like terms, you reduce the number of elements you need to manage, which helps to effectively isolate the variable in the next steps. This technique is commonly used and necessary in algebra to keep equations neat and workable.

Isolating Variables

Isolating the variable is the key step to finding its value in an equation. This process involves performing operations that get the variable on one side of the equation alone. Once it's isolated, you can determine its numerical value.
Starting with the equation after combining like terms:\[33 - 24q = 21 - 27q\]
The aim is to have \(q\) terms on one side. To achieve this:

• Add \(27q\) to both sides to cancel \(-27q\) on the right, giving you: \[33 + 3q = 21\]
• Subtract \(33\) from both sides to move constant terms aside: \[3q = -12\]
• Finally, divide by \(3\) to solve for \(q\), yielding: \[q = -4\]

Each step is intentional to maintain the equation's balance. Remember, whatever operation you do to one side of the equation, you must do to the other to keep it valid. This methodology is applicable to any linear equation you're aiming to solve.