Problem 65 - Algebra 1: Concepts and Skills

Question

Solve the equation. \(6(q+22)=-120\)

Step-by-Step Solution

Answer

\(-42\)

Step 1: Distribute

Distribute \(6\) to the terms inside the parenthesis, resulting in the equation \(6q + 132 = -120\).

Step 2: Simplify by subtracting

To isolate \(q\), subtract \(132\) from both sides to get \(6q = -120 - 132\).

Step 3: Evaluate subtraction

After evaluating the subtraction on the right, the equation becomes \(6q = -252\).

Step 4: Divide

Finally, in order to solve for \(q\), divide both sides by \(6\), leading to the solution \(q = -252 / 6\).

Step 5: Final evaluation

Evaluating the division, the solution for \(q\) is \(q = -42\).

Key Concepts

Distributive Property

The distributive property is a very useful tool in mathematics, especially when dealing with equations. It helps to simplify expressions that involve multiplication over addition or subtraction inside parentheses. In general, the distributive property states that for any numbers or variables, \(a(b + c) = ab + ac\). This means you distribute the multiplication across the terms inside the parenthesis.
In the given original exercise, the equation \(6(q + 22) = -120\) uses the distributive property to eliminate the parentheses by multiplying 6 with both \(q\) and 22, resulting in the expression \(6q + 132\). The next steps in solving the equation become much simpler once the parentheses are removed. This is the starting point when solving many algebraic equations that include a set of parentheses.

Isolating Variables

Isolating the variable is a crucial step in solving equations. To solve an equation for a specific variable means to get that variable by itself on one side of the equation with no other terms. In this exercise, our goal is to isolate \(q\).
After applying the distributive property, you're left with the equation \(6q + 132 = -120\). To begin isolating \(q\), you need to eliminate \(132\) from the left side. This is done by performing the inverse operation, which means subtracting \(132\) from both sides of the equation.

• This subtraction gives you \(6q = -252\).

Now that \(132\) is gone, you have isolated \(6q\) to one side. The next step is to get \(q\) alone by dividing both sides by 6.

Simplifying Expressions

Simplifying expressions is a core skill in algebra. It involves combining like terms and making the equation easier to manage. In this context, simplifying can mean both combining constants and reducing the complexity of the equation.
In our original problem, once the distributive property was applied, we had \(6q + 132\). When subtracting 132 from both sides, a key part of simplifying happens: \(-120 - 132\) is simplified to \(-252\).
The operation of subtraction here not only reduces the complexity of the equation but also helps in isolating the variable \(q\) effectively.
Being consistent in simplifying expressions as early as possible will aid in solving equations more comfortably and correctly.

Linear Equations

Linear equations are equations of the first degree, meaning they have no exponents higher than one. The standard form is \(ax + b = c\), and they graph as straight lines.
In our exercise, \(6q + 132 = -120\) is a linear equation because it can be rearranged into the typical form once simplified. Solving linear equations generally requires isolating the variable through steps like applying the distributive property, simplifying, and then solving by arithmetic operations.

• They are straightforward thanks to their simplicity - just apply arithmetic operations in a sequence to find the value of the variable.
• The solution gives a single value for the variable, which can be represented directly on a number line.

Knowing how to solve linear equations is foundational in algebra and forms the basis for more complex problems as students advance in mathematics.