Problem 17 - Algebra 2

Question

Solve each system of equations. \(\begin{aligned} 9 x-3 y+12 z &=39 \\ 12 x-4 y+16 z &=52 \\ 3 x-8 y+12 z &=23 \end{aligned}\)

Step-by-Step Solution

Answer

Due to identical simplification, this verifies solutions found where overlapping, suggesting parameterized or checking consistent initial overlap.

Step 1: Write the System of Equations

The given system of equations is: 1. \(9x - 3y + 12z = 39\)2. \(12x - 4y + 16z = 52\)3. \(3x - 8y + 12z = 23\).We need to solve for \(x\), \(y\), and \(z\).

Step 2: Simplify Equation 2

Divide the second equation by 4 to simplify it: \(3x - y + 4z = 13\).

Step 3: Subtract Simplified Equation 2 from Equation 1

Take the simplified equation from step 2 and solve:Subtract \(3x - y + 4z = 13\) from \(9x - 3y + 12z = 39\) to eliminate the common terms: \((9x - 3y + 12z) - (3x - y + 4z) = 39 - 13\)This simplifies to: \(6x - 2y + 8z = 26\).Simplify further by dividing the equation by 2: \(3x - y + 4z = 13\).

Step 4: Subtract Equation 1 from Equation 2

Original system gives: Subtract \(12x - 4y + 16z = 52\) from \(9x - 3y + 12z = 39\) to yield:\((12x - 4y + 16z) - (9x - 3y + 12z) = 52 - 39\)This simplifies to \(3x - y + 4z = 13\), which is similar to the simplified Equation 2.

Step 5: Simplify Equation 3

Check Equation 3:Simplify \(3x - 8y + 12z = 23\) using the same approach.Since the previous equations are expressed as \(3x - y + 4z = 13\), we should verify any combination with Equation 3 is consistent.

Step 6: Combine and Solve Equations

Now we will use consistent and simplified forms found:Using the repeated and simplified equation \(3x - y + 4z = 13\), and comparison against Equation 3 check where overlap in terms results in solutions.

Key Concepts

Linear Equations

Linear equations are equations where each term is either a constant or a product of a constant and a single variable. They are the simplest forms of equations in algebra and take the general form \( ax + by + cz = d \).

• Here, \(a\), \(b\), and \(c\) are coefficients, and \(d\) is the constant term.
• Linear equations form straight lines when graphed.

To solve linear equations, we primarily focus on isolating variables to find their values. This involves arranging the equation such that one side is the variable of interest and the other is its calculated value. Linear equations are crucial in solving systems as they provide relationships between variables.

Solving Systems

A system of equations consists of two or more equations that share variables. Solving systems means finding values for the variables that satisfy all equations simultaneously. There are several methods to solve systems:

• Substitution: Solve one equation for one variable, and substitute that expression into the other equations.
• Elimination: Add or subtract equations to eliminate a variable, making it easier to solve for the others.
• Graphical: Graph each equation and identify the point(s) where they intersect.

For systems involving linear equations, the goal is to determine whether the equations intersect at a single point (unique solution), overlap completely (infinite solutions), or are parallel (no solution).

Elimination Method

The elimination method is a technique used to simplify and solve systems of linear equations by removing one variable at a time. Let's break down the steps:

• Align the equations so the variables are in columns.
• Decide which variable to eliminate first by adding or subtracting equations.
• Simplify resulting equations, reducing the system to two variables, then one.
• Solve the remaining equation for one unknown and back-substitute to find others.

In our original exercise, this method was used by subtracting one equation from another to eliminate variables step by step, simplifying complex systems down to manageable parts, making it easier to find the exact values of each unknown.

Algebraic Simplification

Algebraic simplification involves reducing equations to their simplest forms. This process often includes dividing an entire equation by a common factor, combining like terms, and reducing fractions.

• Identify common coefficients or factors to streamline equations.
• Combine like terms to simplify expressions and make calculations more manageable.
• Ensure all terms are unified in one consistent form.

In the exercises, we simplified equations by removing coefficients and reducing fractions to more easily apply further mathematical techniques. Simplification ensures that solving becomes less error-prone and more straightforward, providing clearer solutions to complex systems of equations.