Problem 31 - Precalculus: Mathematical for Calculus

Question

$$\left\\{\begin{aligned} x+y-2 z &=3 \\ 2 x+5 z &=11 \\ 2 x+3 y &=12 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution is \( x = 3 \), \( y = 2 \), \( z = 1 \).

Step 1: Understand the System of Equations

We have a system of three linear equations:1. \( x + y - 2z = 3 \)2. \( 2x + 5z = 11 \)3. \( 2x + 3y = 12 \).We need to solve for the variables \( x \), \( y \), and \( z \) that satisfy all these equations simultaneously.

Step 2: Express x in Terms of z

From the second equation, \( 2x + 5z = 11 \), we can express \( x \) in terms of \( z \):\[ 2x = 11 - 5z \]\[ x = \frac{11 - 5z}{2} \].

Step 3: Substitute x in the Third Equation

Substitute \( x = \frac{11 - 5z}{2} \) into the third equation \( 2x + 3y = 12 \):\[ 2\left(\frac{11 - 5z}{2}\right) + 3y = 12 \]\[ 11 - 5z + 3y = 12 \]\[ 3y = 12 - 11 + 5z \]\[ 3y = 1 + 5z \]\[ y = \frac{1 + 5z}{3} \].

Step 4: Substitute x and y in the First Equation

Substitute \( x = \frac{11 - 5z}{2} \) and \( y = \frac{1 + 5z}{3} \) into the first equation \( x + y - 2z = 3 \):\[ \frac{11 - 5z}{2} + \frac{1 + 5z}{3} - 2z = 3 \].

Step 5: Solve for z

Find a common denominator (6) and solve for \( z \):\[ \frac{3(11 - 5z) + 2(1 + 5z)}{6} - 2z = 3 \]\[ \frac{33 - 15z + 2 + 10z}{6} - 2z = 3 \]\[ \frac{35 - 5z}{6} = 3 + 2z \]Multiply both sides by 6:\[ 35 - 5z = 18 + 12z \]\[ 35 - 18 = 12z + 5z \]\[ 17 = 17z \]\[ z = 1 \].

Step 6: Calculate x and y

Use \( z = 1 \) to find \( x \) and \( y \):From \( x = \frac{11 - 5z}{2} \):\( x = \frac{11 - 5(1)}{2} = \frac{6}{2} = 3 \).From \( y = \frac{1 + 5z}{3} \):\( y = \frac{1 + 5(1)}{3} = \frac{6}{3} = 2 \).

Step 7: Verify Solutions

Check solutions in all equations:1. \( 3 + 2 - 2(1) = 3 \): True2. \( 2(3) + 5(1) = 6 + 5 = 11 \): True3. \( 2(3) + 3(2) = 6 + 6 = 12 \): TrueAll equations are satisfied.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that describe straight lines on a graph. They typically take the form \( ax + by + cz = d \), where \( a \), \( b \), and \( c \) are coefficients, and \( d \) is a constant. These equations can have one or more variables. In our system of equations:

• Equation 1: \( x + y - 2z = 3 \)
• Equation 2: \( 2x + 5z = 11 \)
• Equation 3: \( 2x + 3y = 12 \)

Each equation uses the variables \( x \), \( y \), and \( z \). Our goal is to find values for these variables that make all three equations true simultaneously. Linear equations are foundational in algebra as they help model a wide array of real-world problems.

Variable Substitution

Variable substitution is a technique used to simplify solving a system of equations. It involves replacing one variable with an expression derived from another equation. This substitution makes the system easier to solve by reducing the number of variables in an equation.

Step-by-Step Substitution

We start by expressing one variable in terms of another. From the second equation, \( 2x + 5z = 11 \), we can isolate \( x \):
\[ x = \frac{11 - 5z}{2} \]
Next, substitute this expression for \( x \) in the third equation, \( 2x + 3y = 12 \):
\[ 2\left(\frac{11 - 5z}{2}\right) + 3y = 12 \]
This simplifies to \( y = \frac{1 + 5z}{3} \). By substituting the expressions for \( x \) and \( y \) into the first equation, we systematically reduce the complexity.

Solution Verification

Verifying solutions in problems involving systems of equations means checking if the solutions satisfy each original equation. This ensures there are no calculation errors and the values are correct.

• For Equation 1: Substitute \( x = 3 \), \( y = 2 \), and \( z = 1 \) into the equation \( x + y - 2z = 3 \). It simplifies to \( 3 + 2 - 2(1) = 3 \), which holds true.
• For Equation 2: Substitute into \( 2x + 5z = 11 \): \( 2(3) + 5(1) = 11 \). The calculations show our solution is correct.
• For Equation 3: Substitute into \( 2x + 3y = 12 \): \( 2(3) + 3(2) = 12 \), confirming the solution works.

Each equation is satisfied, confirming that \( x = 3 \), \( y = 2 \), and \( z = 1 \) is indeed the solution to the system.

Expressing Variables

Expressing variables is a common technique used to revise the structure of an equation, making it easier to solve. This is done by isolating one particular variable and expressing it in terms of other variables and constants. For example, in the equation \( 2x + 5z = 11 \), we solved for \( x \):
\[ x = \frac{11 - 5z}{2} \]
This transformation is key as it helps reduce the number of unknowns to solve systematically. It's used in conjunction with substitution to narrow down possible solutions step by step. Meanwhile, writing \( y = \frac{1 + 5z}{3} \) from \( 3y = 1 + 5z \) demonstrates this principle effectively. Expressing variables works as a roadmap through a system of linear equations, guiding us toward the correct solutions.