Problem 60 - Algebra 1: Concepts and Skills

Question

Use mental math to fill in the missing number so that all the equations have the number 6 as a solution. a. \(?+x=18\) b. \(\quad ? \quad x=30 \quad\) c. \(\frac{?}{x}=6\)

Step-by-Step Solution

Answer

The missing numbers in the equations are 12, 5, and 36 respectively.

Step 1: Solve for the unknown in the first equation

Since the equation is given by \( ? + x = 18 \) and we know that the solution should be 6, we can substitute 6 for x. This gives us: \( ? + 6 = 18 \). Solving for ? gives us ? = 18 - 6 which equals 12.

Step 2: Solve for the unknown in the second equation

The equation is \( ? \times x = 30 \). We know that the solution should be 6 hence we substitute 6 for x, giving us: \( ? \times 6 = 30 \). To solve for ?, we divide both sides by 6. Therefore, ? = 30 ÷ 6 equals 5.

Step 3: Solve for the unknown in the last equation

The equation given is \(\frac{?}{x} = 6\). Again, substituting 6 for x, we get: \(\frac{?}{6} = 6\). To solve for ?, we multiply both sides by 6; consequently, ? = 6 \times 6 which equals 36.

Key Concepts

Mental Math

Mental math refers to performing calculations in your head without the assistance of external devices or tools. It improves your ability to think quickly and solve mathematical problems efficiently. In the given exercise, mental math can be applied to simplify solving for ?, by using basic arithmetic operations. Consider the first part of the exercise:

• For the equation \(? + x = 18\), replace \(x\) with 6, making it \(? + 6 = 18\). You can quickly subtract 6 from 18 in your head to find that \(?\) is 12.

The ability to handle such operations mentally enhances your problem-solving skills and speed. Practicing mental math regularly helps you become adept at tackling more complex algebraic equations.

Solving for Unknowns

Solving for unknowns involves finding the value of an unknown variable in an equation. This skill is crucial in algebra and forms the basis for more advanced math problems. In the exercise, the unknown is represented by a question mark because its value isn't provided.

• For \(? \times x = 30\), replace \(x\) with 6 resulting in \(? \times 6 = 30\). To solve for \(?\), divide both sides by 6 to get \(? = 5\).

Understanding how to manipulate equations to isolate the unknown variable is key. Make sure to practice different types of equations to reinforce these concepts. It's all about balancing the equation to find the hidden value.

Substitution Method

The substitution method is a straightforward way to solve equations by replacing one variable with a given value. Here, you substitute 6 for \(x\) in all equations, reflecting the solution provided in the exercise.

• In \(\frac{?}{x} = 6\), replace \(x\) with 6, leading to \(\frac{?}{6} = 6\). To isolate \(?\), multiply both sides by 6 resulting in \(? = 36\).

This method is particularly useful when an equation has one variable. It simplifies the solving process by reducing it to a basic arithmetic operation. Substitution helps build a strong foundation for tackling more complex equations in algebra.