Problem 61 - Algebra for College Students

Question

For Problems \(61-80\), solve each problem by using a system of equations. The sum of two numbers is 53 , and their difference is 19. Find the numbers.

Step-by-Step Solution

Answer

The numbers are 36 and 17.

Step 1: Define the Variables

Let the two numbers be \(x\) and \(y\).

Step 2: Set Up the Equations

According to the problem, first equation is the sum of the numbers: \(x + y = 53\). The second equation is the difference of the numbers: \(x - y = 19\).

Step 3: Solve the System by Addition

To eliminate \(y\), add the two equations together: \((x + y) + (x - y) = 53 + 19\). This simplifies to \(2x = 72\).

Step 4: Solve for \(x\)

Divide both sides by 2 to find \(x\): \(x = \frac{72}{2} = 36\).

Step 5: Solve for \(y\)

Substitute \(x = 36\) into the first equation: \(36 + y = 53\). Solve for \(y\): \(y = 53 - 36 = 17\).

Step 6: Verify the Solution

Check the solution with both original equations. Indeed, \(36 + 17 = 53\) and \(36 - 17 = 19\), confirming the solution is correct.

Key Concepts

Solving Linear Equations

In the realm of algebra, solving linear equations is a foundational skill. A linear equation is essentially an equation where each term is either a constant or the product of a constant and a single variable. These equations can represent straight lines when graphed. In our exercise, we had two linear equations:

• \(x + y = 53\)
• \(x - y = 19\)

These equations are termed as "linear" because they graph as straight lines. By solving a system of these two equations, we were able to find the values of the variables \(x\) and \(y\). The solution is where these two lines intersect on a graph.


To solve these equations, we use the method of addition. Adding equations is a powerful technique because it allows us to eliminate one variable, making it easier to solve for the other variable. By combining \(x + y = 53\) and \(x - y = 19\), we managed to get \(2x = 72\), thereby eliminating \(y\) and allowing us to solve for \(x\). Once we knew \(x\), finding \(y\) was straightforward by substituting \(x\) back into one of the initial equations.

Algebraic Problem Solving

Algebraic problem solving involves breaking down a math problem into its basic elements. It requires setting up equations based on the problem's conditions and solving them step-by-step. For instance, in our problem, we first defined our variables, \(x\) and \(y\), to represent the two unknown numbers we seek.

• The sum of these numbers was given, leading to the equation \(x + y = 53\).
• Their difference was provided, forming the equation \(x - y = 19\).

Creating these equations is an example of translating a word problem into a mathematical form. This is a crucial step in algebraic problem-solving, where assumptions from the problem statement are used to form equations.

After forming the equations, solving them involves using algebraic rules and techniques, such as addition, subtraction, multiplication, or division, until the unknowns are found. In this problem, the method of elimination helped us to isolate one of the variables, allowing us to find exact values for both numbers.

Mathematical Reasoning

Mathematical reasoning is the glue that holds algebra and problem-solving together. It is the ability to analyze a problem, break it down into manageable parts, and methodically solve each part. In our particular problem, mathematical reasoning began with understanding what the problem was asking.

We knew that:

• Two numbers must sum up to 53.
• These numbers must have a difference of 19.

These bits of information led to the formation of our two equations. Reasoning about the best way to solve these equations, we decided to add them to eliminate \(y\). This strategic decision was vital in simplifying the system of equations.

Finally, mathematical reasoning involves verifying our solution. We checked our answers by substituting \(x = 36\) and \(y = 17\) back into the original equations. Seeing that both conditions held true confirmed the accuracy of our results. Thus, mathematical reasoning was key not only in finding the solution but also in ensuring that it was correct.