Problem 32 - Elementary Algebra

Question

In the following exercises, solve each number word problem. The sum of two numbers is -284. One number is 62 less than the other. Find the numbers.

Step-by-Step Solution

Answer

The numbers are -111 and -173.

Step 1 - Define Variables

Let's define the variables first. Let the first number be denoted by \(x\) and the second number be denoted by \(y\).

Step 2 - Set Up Equations

Based on the problem, we have two pieces of information:1. The sum of the two numbers is -284: \(x + y = -284\).2. One number is 62 less than the other: \(y = x - 62\).

Step 3 - Substitute the Second Equation into the First

Substitute \(y = x - 62\) into the first equation: \[x + (x - 62) = -284\].

Step 4 - Solve for x

Combine like terms and solve for \(x\):\[x + x - 62 = -284\] \[2x - 62 = -284\] \[2x = -284 + 62\] \[2x = -222\] \[x = -111\].

Step 5 - Solve for y

Use the value of \(x\) to find \(y\): \[y = x - 62\] \[y = -111 - 62\] \[y = -173\].

Step 6 - State the Final Answer

The two numbers are \(x = -111\) and \(y = -173\).

Key Concepts

Algebraic Equations

When solving number word problems, we often use algebraic equations. An algebraic equation is a mathematical statement that shows the relationship between different quantities.
These relationships are expressed using variables (like x and y) and mathematical operations (like addition and subtraction).
In the example problem, we created two equations based on the information given:
1. The sum of the two numbers is -284 ( x + y = -284 ).
2. One number is 62 less than the other ( y = x - 62 ).
These equations help us understand how the unknown numbers relate to each other and set the stage for finding their values.

Variable Substitution

Variable substitution is a technique where we replace one variable with an expression involving another variable.
This allows us to simplify the problem and solve for one unknown at a time.
In the given problem, we substitute the second equation ( y = x - 62 ) into the first equation ( x + y = -284 ).
Instead of dealing with both x and y, we're now working with just one variable:
x + (x - 62) = -284 .
This simplification makes it easier to solve the equation and find the value of x, which we can then use to find y.

Solving for Variables

Once we have our simplified equation, we need to solve for the variable. This involves isolating the variable on one side of the equation.
In the example problem, we simplify x + (x - 62) = -284 to 2x - 62 = -284 .
Our goal is to isolate x. Here’s a step-by-step breakdown:
1. Combine like terms: x + x becomes 2x .
2. Move constants to the other side by adding or subtracting: 2x - 62 = -284 becomes 2x = -284 + 62 .
3. Perform the arithmetic: 2x = -222 .
4. Isolate x by dividing by 2: x = -111 .
With x solved, we substitute it back into one of our original equations to find y: y = x - 62 becomes y = -111 - 62 , resulting in y = -173 .