Problem 43 - Algebra and Trigonometry

Question

Find two numbers whose sum is 34 and whose difference is 10.

Step-by-Step Solution

Answer

The two numbers are 22 and 12.

Step 1: Set Up Variables for the Numbers

Let's call the two numbers we want to find \( x \) and \( y \). We know they have a sum of 34 and a difference of 10.

Step 2: Write the System of Equations

Based on the problem, we can form a system of equations: \[ x + y = 34 \] and \[ x - y = 10 \].

Step 3: Add the Equations

To eliminate one of the variables, add the two equations: \( (x + y) + (x - y) = 34 + 10 \). This simplifies to \( 2x = 44 \).

Step 4: Solve for x

Divide both sides of the equation \( 2x = 44 \) by 2 to find \( x \). Thus, \( x = \frac{44}{2} = 22 \).

Step 5: Substitute x into One Equation

Now that we know \( x = 22 \), substitute this value into the sum equation: \( x + y = 34 \) becomes \( 22 + y = 34 \).

Step 6: Solve for y

Subtract 22 from both sides of the equation \( 22 + y = 34 \) to find \( y \). So, \( y = 34 - 22 = 12 \).

Step 7: Verify the Solution

Check the solution by substituting \( x = 22 \) and \( y = 12 \) into the original equations. The sum \( x + y = 22 + 12 = 34 \) and the difference \( x - y = 22 - 12 = 10 \) are both correct.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that form a straight line when graphed. They are polynomial equations of degree one, meaning they involve variables raised to the first power. In a typical linear equation, you'll often see forms such as \( ax + b = 0 \), where \( a \) and \( b \) are constants. Usually, the objective is to find the value(s) of the variable \( x \) that make the equation true.

Linear equations are simple yet powerful tools in algebra that help us model relationships where the rate of change is constant. For example, determining the relationship between price and quantity, distance over time, or, as in our problem, figuring out unknown numbers based on given conditions like their sum and difference.

To solve a linear equation, we typically perform operations such as addition, subtraction, multiplication, or division.

• These operations help isolate the variable, making it easy to find its value and thus solve the equation perfectly.
• The linearity makes it straightforward to graph and understand visually if needed.

Substitution Method

The substitution method is a popular strategy used to solve systems of equations, such as those involving linear equations. This technique involves solving one equation for a single variable and substituting that expression into the other equation. It effectively reduces the system to a single equation with one variable.

In our exercise, we were given two linear equations formed from the sum and the difference of two numbers:

• \( x + y = 34 \)
• \( x - y = 10 \)

The steps to apply the substitution method are:

First, add the two equations to eliminate one variable. This allowed us to solve for \( x \) directly:
\( 2x = 44 \) leads to \( x = 22 \).

Next, substitute the value of \( x \) back into one of the original equations to solve for \( y \). By substituting \( x = 22 \) into \( x + y = 34 \), we simplify and solve \( y = 12 \).

This method is particularly useful when one equation can be easily manipulated, making the procedure smooth and efficient.

Verification of Solutions

Verification is the crucial step of checking whether the proposed solutions satisfy all the given conditions of the problem. This ensures the accuracy of your solutions and confirms that no calculation errors were made during the process.

For our problem, we found \( x = 22 \) and \( y = 12 \). To verify:

• Check the sum: Substitute both \( x \) and \( y \) into the equation \( x + y = 34 \).
• Result: \( 22 + 12 = 34 \), which matches the sum condition.

Next:

• Check the difference: Substitute both \( x \) and \( y \) into the equation \( x - y = 10 \).
• Result: \( 22 - 12 = 10 \), confirming the difference condition is met.

Through this process, we can confidently assert that the found values for \( x \) and \( y \) are indeed the correct solutions. Verification acts as a reliable checker ensuring the method used is correct, and the values meet all the problem's requirements.