Problem 40 - Algebra for College Students

Question

Solve each problem. Ted Williams and Rogers Hornsby were two great hitters in Major League Baseball. Together, they had 5584 hits in their careers. Hornsby had 276 more hits than Williams. How many hits did each have?

Step-by-Step Solution

Answer

Ted Williams had 2654 hits, and Rogers Hornsby had 2930 hits.

Step 1: Identify the Variables

Let’s use variables to represent the number of hits for each player. Let W represent the number of hits Ted Williams had, and let H represent the number of hits Rogers Hornsby had.

Step 2: Set Up the Equations

Based on the problem, we know two things: 1. Together, they had 5584 hits, which gives us the equation: \[ W + H = 5584 \] 2. Hornsby had 276 more hits than Williams, which gives us another equation: \[ H = W + 276 \]

Step 3: Substitute the Second Equation into the First

To solve for W, replace H in the first equation with the expression from the second equation: \[ W + (W + 276) = 5584 \]

Step 4: Combine Like Terms and Solve for W

Combine the W terms on the left side of the equation: \[ 2W + 276 = 5584 \] Now, subtract 276 from both sides: \[ 2W = 5308 \] Next, divide both sides by 2: \[ W = 2654 \]

Step 5: Solve for H

Using the value of W found (2654), substitute it back into the equation for H: \[ H = W + 276 = 2654 + 276 = 2930 \]

Step 6: Verify the Solution

Check if the sum of hits equals 5584: \[ W + H = 2654 + 2930 = 5584 \] Both conditions are satisfied.

Key Concepts

Variable Identification

When solving word problems involving unknown quantities, the first crucial step is to identify the variables. Variables are symbols that represent quantities we don't yet know but want to find. In our exercise, we needed to determine the number of hits for two baseball players: Ted Williams and Rogers Hornsby.

To do this, we assign a variable to each player’s hits:

• Let W represent the number of hits Ted Williams had.
• Let H represent the number of hits Rogers Hornsby had.

This setup allows us to transform the word problem into mathematical equations, making it easier to solve.

Always read the problem carefully to pinpoint exactly what you need to find and ensure that each piece of information is accounted for. This helps prevent errors and confusion later in the process.

Substitution Method

The substitution method is an algebraic technique used to solve systems of linear equations. It involves solving one equation for one variable and then substituting that expression into the other equation.

In our exercise, we had the following equations:

• W + H = 5584
• H = W + 276

Here’s how the substitution method works:

• First, solve the second equation for H: H = W + 276
• Next, substitute this expression for H into the first equation: W + (W + 276) = 5584

This substitution substitutes the variable H with the expression involving W; thus, turning a system of two equations into a single equation with one variable, making it simpler to solve. Once you solve the new equation, plug the value back into the original equations to find the other variable.

Linear Equations in Two Variables

A linear equation in two variables is an equation of the form ax + by = c, where x and y are the variables, and a, b, and c are constants. These equations can represent real-world situations and are essential in various fields such as science, engineering, and economics.

In our problem, the two equations:

• W + H = 5584
• H = W + 276

are both linear equations in two variables (W and H). Such equations can graph as straight lines on a coordinate plane, and their intersection represents the solution to the system.

Solving these equations often involves methods like substitution or elimination, helping us find where the two lines intersect. Understanding how to set up and solve linear equations prepares you for more complex mathematical concepts and real-life problem-solving. Always double-check your work to ensure that the solutions satisfy all given conditions.