Problem 30 - Elementary Algebra

Question

In the following exercises, solve each number word problem. The sum of two numbers is -61. One number is 35 more than the other. Find the numbers.

Step-by-Step Solution

Answer

The numbers are -13 and -48.

Step 1: Define Variables

Let the two numbers be represented as x and y. According to the problem, the sum of these two numbers is -61, and one number is 35 more than the other. We can set up the following equations: 1) x + y = -61 2) x = y + 35

Step 2: Substitute the Second Equation Into the First

Substitute the expression for x from the second equation into the first equation. This gives us: y + 35 + y = -61

Step 3: Simplify and Solve for y

Combine like terms and solve for y: 2y + 35 = -61 Subtract 35 from both sides: 2y = -96 Divide both sides by 2: y = -48

Step 4: Solve for x

Using the value of y, substitute back into the second equation to find x: x = y + 35 x = -48 + 35 x = -13

Step 5: Verify the Solution

Verify by substituting the values back into the original equations to ensure they are correct: 1) x + y = -61 -13 + (-48) = -61 (True) 2) x = y + 35 -13 = -48 + 35 (True) The solutions are correct.

Key Concepts

Word Problems

Word problems in algebra are essentially real-life scenarios presented in a mathematical format. They require you to set up and solve equations based on the given information. In our exercise, we need to find two numbers with specific relationships: their sum is -61, and one number is 35 more than the other. These types of problems often involve identifying what the numbers or variables represent and then forming equations to solve them.

Linear Equations

Linear equations are the backbone of algebraic word problems. These are equations of the first degree, meaning they involve variables raised only to the power of one. In our problem, the linear equations are:

• x + y = -61
• x = y + 35

)These equations help us establish the relationship between the two unknown numbers. By solving these equations, we find the exact values of the variables x and y. Linear equations are generally straightforward and involve simple arithmetic operations like addition, subtraction, multiplication, and division.

Variable Substitution

Variable substitution is a crucial technique in algebra to simplify and solve equations that contain multiple variables. In this exercise, we have two variables x and y. The second equation x = y + 35 is substituted into the first equation x + y = -61. This process simplifies the problem from two variables down to one:\[ y + 35 + y = -61 \]Now we can combine the y terms, solve for y, and then use that value to find x. This method reduces the complexity of solving systems of equations and makes it easier to find the solution.

Verification of Solutions

Verification of solutions is the final step to ensure that our answers satisfy all given conditions in the problem. For this exercise, we found that y = -48 and x = -13. To verify, we substitute these values back into the original equations:

• x + y = -61
• -13 + (-48) = -61(True)
• x = y + 35
• -13 = -48 + 35(True)

Both equations are true, confirming that our solutions are correct. Verification is a crucial step to avoid mistakes and ensure accuracy in mathematical problem-solving.