Problem 44 - Algebra 1: Concepts and Skills

Question

SOLVING WITH MENTAL MATH Use mental math to solve the equation. $$ 29-d=10 $$

Step-by-Step Solution

Answer

The solution to the equation is \(d=19\).

Step 1: Analyze the given equation

The equation given is \(29-d=10\). The goal is to find the value of 'd'. To do this, 'd' needs to be isolated on one side of the equation.

Step 2: Rearrange the equation to isolate 'd'

To do this, the constant value from the left hand side needs to be moved to the right hand side of the equation. In this equation, the 'd' is subtracted from 29. It can be moved to the other side of the equation by adding 'd' to both sides resulting in \(29=10+d\).

Step 3: Solve for 'd'

Now, we subtract 10 from both sides of the equation to isolate 'd' resulting in the solution \(d=29-10\). Using basic subtraction, we can find that 'd' is equal to 19.

Key Concepts

Solving Equations

Solving equations is like finding the answer to a puzzle. An equation is essentially a question about what numbers can make a statement true. In our exercise, we are given a puzzle to solve: \(29-d=10\). To tackle this using mental math, it's important to remember the equation's goal: make sure both sides are equal. This means, if you find the right value of \(d\), you will ensure that the total value on the left side equals the number on the right side.Here is a step-by-step approach to solving equations with mental math:

• Look at the equation and identify the variable you need to solve for, like \(d\) here.
• Think about what operations are performed on that variable. If something is subtracted, added, multiplied, or divided, you’ll consider doing the opposite to simplify the equation.
• Apply this operation in your head until you find a balance—that's your answer!

Solving equations helps develop logical thinking and problem-solving skills.

Basic Subtraction

Subtraction is a fundamental math operation that tells you how much is left when you take one number away from another. In the equation \(29-d=10\), subtraction plays a key role. Learning to understand subtraction fully is crucial for mental math.Here's how to boil subtraction down:

• "What is left?" This question leads you to perform the subtraction.
• When you subtract, you remove value. Like asking, "if I have 29, and I subtract \(d\), I am left with 10. What must \(d\) be to make this statement true?"
• Understand simple terms: "take away" or "less six" (for example, \(29 \text{ take away } 19 = 10\)).

Subtraction forms the basis of understanding how math can manipulate numbers to find missing values.

Isolating Variables

Isolating a variable means separating it on one side of the equation. In the equation \(29-d=10\), the goal was to isolate \(d\). This process helps in making the equation simpler to solve.Here's how you can think of isolating variables:

• Identify the variable you want to isolate. In our equation, it's \(d\).
• Use opposite operations to eliminate other numbers from this side. If something is subtracted from your variable, you add it to both sides of the equation.
• Continue until the variable stands alone. For example, moving \(-d\) to the right side allows you to rewrite the equation as \(29 = 10 + d\).
• Finally, simplify the equation to determine the value of the variable, like finding \(d=19\) by subtracting 10 from 29.

Isolating variables teaches you the essence of balance in equations and helps harness the power of algebra in solving math problems.