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Chapter 2: Problem 55

Translate to an equation and then solve it. The sum of \(y\) and -30 is 40 .

Short Answer

Expert verified
y = 70

Step by step solution

01

Understand the problem

We are given that the sum of a number, which we'll call \(y\), and -30 is 40.
02

Translate the problem into an equation

The phrase 'the sum of \(y\) and -30 is 40' can be written as an equation: \( y + (-30) = 40 \)
03

Simplify the equation

Rewrite the equation to a more straightforward form: \( y - 30 = 40 \)
04

Solve for \(y\)

Add 30 to both sides of the equation to isolate \(y\): \[ y - 30 + 30 = 40 + 30 \] \[ y = 70 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

equation translation
When solving word problems, the first crucial step is to translate the written statement into a mathematical equation. This process is called equation translation. Let's break down the given problem: 'The sum of y and -30 is 40.'

The word 'sum' indicates addition. Therefore, the sum of a number (which we call y) and -30 can be written as:

\( y + (-30) \)

The problem further tells us that this sum equals 40. Hence, we can write it as an equation:

\( y + (-30) = 40 \)

Translating words into equations is key to solving algebraic problems. Once you have the equation, the next steps become easier.
simplification
In mathematics, simplification involves rewriting an equation or expression to its simplest form. This can help in clearly understanding and solving the problem.

Let's take the equation we translated: \( y + (-30) = 40 \)

To simplify the equation, we can rewrite the combination of y and -30 in a more straightforward form. Adding a negative number is the same as subtracting the positive version of that number. Thus, adding -30 is the same as subtracting 30.

Therefore, we can rewrite:

\( y + (-30) \text{ as } y - 30 \)

This gives us the simplified equation: \( y - 30 = 40 \)

Simplifying the equation makes it easier to perform further arithmetic operations.
isolation of variables
The final step in solving a linear equation is isolating the variable, meaning getting the variable y by itself on one side of the equation.

From our simplification, we have: \( y - 30 = 40 \)

To isolate y, we need to eliminate the -30 term from the left side of the equation. We can do this by performing the inverse operation, which is addition. We add 30 to both sides of the equation:

\[ y - 30 + 30 = 40 + 30 \]

Simplifying this, we get: \[ y = 70 \]

Now, y is isolated, and we've found that y equals 70.

Isolating the variable is a core step in solving equations, allowing us to find the solution effectively.

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