Problem 77 - Algebra 1

Question

Write a question that can be represented by the equation. Then use mental math to solve the equation. $$y-7=3$$

Step-by-Step Solution

Answer

A suitable question could be 'What number minus 7 equals 3?'. The solution to the equation \(y-7=3\) using mental math is \(y = 10\).

Step 1: Formulate an appropriate question to represent the equation

It is needed to write a question that corresponds the equation \(y-7=3\). One could be: What number minus 7 equals 3?

Step 2: Solve the equation using mental math

The equation \(y-7=3\) can be rewritten as \(y = 3 + 7\) to make it easier to solve mentally. By simply adding 3 to 7, the answer for \(y\) is found to be 10.

Key Concepts

Solving Equations

Solving equations is a fundamental concept in basic algebra that involves finding the value of a variable that makes an equation true. An equation resembles a balance scale, where each side must equal the other. In the example provided, the equation is \( y - 7 = 3 \). The goal is to find the value of \( y \) that makes the left-hand side equal to the right-hand side.

To solve this, we need to isolate the variable \( y \). This can be done by reversing the operation that is currently affecting \( y \). Since \( y \) is being decreased by 7, we do the opposite by adding 7 to both sides of the equation. This gives:
\[ y - 7 + 7 = 3 + 7 \]
Where the minus 7 and plus 7 cancel each other out, simplifying to \( y = 10 \). This is our solution, which indicates that \( y \) equals 10 will satisfy the original equation.

It is key to remember that any operation you perform on one side must be performed on the other to maintain the balance of the equation.

Mental Math

Mental math refers to the practice of doing calculations in your head, without the use of calculators or paper. It sharpens numerical intuition and helps in solving equations quickly.

For the equation \( y - 7 = 3 \), you can use mental math to identify the value of \( y \). Begin by considering what number you need to subtract 7 from to get 3. By understanding the relationship between addition and subtraction, you recognize that adding 7 to 3 gets you back to the original number, hence:
\[ 3 + 7 = 10 \]
This mental manipulation allows you to quickly and effectively derive the solution without writing anything down.

Practicing mental math develops confidence and skills that can be extremely useful during exams or any situation when tools are not available. By continuously performing small calculations mentally, you'll improve your speed and accuracy.

Equation Formulation

Equation formulation is the process of translating a real-world problem or statement into a mathematical equation, which can then be solved to find an answer. This skill is crucial in making connections between mathematical symbols and their practical applications.

In the given exercise, a mathematical sentence is initially crafted: "What number minus 7 equals 3?" This question directly corresponds to the equation \( y - 7 = 3 \). The word 'what number' indicates the variable \( y \), and the operations of subtraction are reflected in the equation.

Formulating equations helps in simplifying complex scenarios into manageable mathematical problems. It is also an important step in developing mathematical problem-solving skills as it involves understanding the problem context and translating it precisely into a mathematical form.

When you encounter descriptive problems, practice identifying key phrases and operations that can be converted into symbols and numbers. This habit will speed up your ability to form and solve equations effectively.