Problem 61 - Intermediate Algebra

Question

NBA Records. The three highest one-game point totals by one player in a National Basketball Association game are shown below. Solve the following system to find \(x, y,\) and \(z\) $$ \left\\{\begin{array}{l} x+y+z=259 \\ x-y=19 \\ x-z=22 \end{array}\right. $$ $$ \begin{array}{|c|l|c|} \hline \text { Pts } & \text { Player, team } & \text { Date } \\ \hline x & \text { Wilt Chamberlain, Philadelphia } & 3 / 2 / 1962 \\ y & \text { Kobe Bryant, Los Angeles } & 1 / 22 / 2006 \\ z & \text { Wilt Chamberlain, Philadelphia } & 12 / 8 / 1961 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The values are \(x = 100\), \(y = 81\), \(z = 78\).

Step 1: Set Up Equations

We have a system of equations to solve:\[ \begin{align*} x + y + z &= 259 \ x - y &= 19 \ x - z &= 22 \ \end{align*} \] These equations are derived from the given player stats.

Step 2: Solve for \(y\) from the Second Equation

From \(x - y = 19\), we can isolate \(y\) as follows:\[ y = x - 19 \] Substitute this expression for \(y\) in other equations if needed.

Step 3: Solve for \(z\) from the Third Equation

Similarly, from \(x - z = 22\), isolate \(z\):\[ z = x - 22 \] Substitute this expression for \(z\) in other equations if needed.

Step 4: Substitute \(y\) and \(z\) into the First Equation

Using \(x + y + z = 259\), substitute the expressions for \(y\) and \(z\):\[ \begin{align*} x + (x - 19) + (x - 22) &= 259 \ \end{align*} \] Simplify and solve for \(x\).

Step 5: Simplify and Solve for \(x\)

Combine like terms from \(x + (x - 19) + (x - 22) = 259\): \[ 3x - 41 = 259 \] Add 41 to both sides: \[ 3x = 300 \] Divide by 3: \[ x = 100 \]

Step 6: Find \(y\) and \(z\)

Substitute \(x = 100\) back into the expressions for \(y\) and \(z\):\[ \begin{align*} y &= 100 - 19 = 81 \ z &= 100 - 22 = 78 \ \end{align*} \] These are the values for \(y\) and \(z\).

Step 7: Verify the Solution

Verify the solution by substituting \(x = 100\), \(y = 81\), and \(z = 78\) back into the original equations to ensure they hold true for all.

Key Concepts

System of Equations

In mathematics, a system of equations refers to a collection of two or more equations with a common set of variables. In our example, we are dealing with a system of three linear equations:

• \( x + y + z = 259 \)
• \( x - y = 19 \)
• \( x - z = 22 \)

In these equations, each represents a relationship between three unknowns: \( x \), \( y \), and \( z \). Solving the system means finding values for these variables that satisfy all the equations simultaneously.
Systems of equations can often be solved using several methods, including graphing, substitution, and elimination. They play a crucial role in mathematical modeling, allowing us to express and solve real-world problems where multiple conditions must be satisfied at the same time.

Substitution Method

The substitution method is a popular algebraic technique used to solve systems of equations. It involves isolating one variable in one equation and then substituting this expression into the other equations. This method gradually reduces the system's complexity by breaking down the problems into simpler expressions.To illustrate, in our system, we isolated \( y \) and \( z \) from two of the equations:

• \( y = x - 19 \)
• \( z = x - 22 \)

We then substituted these into the first equation \( x + y + z = 259 \), resulting in an equation with just one variable, \( x \), which we could solve directly.
By using substitution, we make the equations easier to handle, allowing us to systematically work towards finding the solution. This method is especially useful when solving systems where one equation is simple to solve for a particular variable.

Solving Equations

Solving equations is a fundamental process in algebra, aimed at finding values for variables that make the equation true. In the context of our problem, after performing substitution, we ended up with a single equation:

• \( 3x - 41 = 259 \)

The goal is to isolate \( x \) on one side of the equation to find its value. This involves:

• First, simplifying the equation by combining like terms.
• Next, performing basic operations to isolate \( x \), such as addition or subtraction.
• Finally, division to solve for \( x \).

Once \( x \) is found, the values for the other variables, \( y \) and \( z \), are determined by substituting \( x \) back into their respective expressions.
Being comfortable with solving equations not only helps in understanding algebra but also in applying mathematical reasoning to various scientific, engineering, and everyday problems.