Problem 1 - Algebra A Combined Function

Question

Solve each system. $$ \left\\{\begin{aligned} x-y+z &=-4 \\ 3 x+2 y-z &=5 \\ -2 x+3 y-z &=15 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The solution is \( x = -1 \), \( y = 5 \), \( z = 2 \).

Step 1: Label the Equations

We will first label each equation for ease of reference:Equation 1: \( x - y + z = -4 \)Equation 2: \( 3x + 2y - z = 5 \)Equation 3: \( -2x + 3y - z = 15 \).

Step 2: Eliminate z from Equations

To eliminate \(z\), we will add Equations 1 and 2:\( (x - y + z) + (3x + 2y - z) = -4 + 5 \)This simplifies to:\( 4x + y = 1 \). Now label it as Equation 4.

Step 3: Eliminate z from Other Two Equations

Add Equations 1 and 3 to eliminate \(z\):\( (x - y + z) + (-2x + 3y - z) = -4 + 15 \)This simplifies to:\( -x + 2y = 11 \). Now label it as Equation 5.

Step 4: Solve the System of Two Equations

We solve Equations 4 and 5:Equation 4: \( 4x + y = 1 \)Equation 5: \( -x + 2y = 11 \).First, solve Equation 4 for \(y\):\( y = 1 - 4x \).Substitute \(y = 1 - 4x\) into Equation 5:\( -x + 2(1 - 4x) = 11 \).This simplifies to:\( -x + 2 - 8x = 11 \) or \( -9x = 9 \).Thus, \( x = -1 \).

Step 5: Solve for y

Substitute \( x = -1 \) back into \( y = 1 - 4x \):\( y = 1 - 4(-1) = 1 + 4 = 5 \).

Step 6: Solve for z

Using the values \( x = -1 \) and \( y = 5 \) in Equation 1:\( x - y + z = -4 \), substitute \( x \) and \( y \):\( -1 - 5 + z = -4 \).Solve for \( z \): \( z = 2 \).

Step 7: Verify the Solution

Substitute \( x = -1 \), \( y = 5 \), \( z = 2 \) back into the original equations to verify:Equation 1: \( -1 - 5 + 2 = -4 \) is correct.Equation 2: \( 3(-1) + 2(5) - 2 = 5 \) is correct.Equation 3: \( -2(-1) + 3(5) - 2 = 15 \) is correct.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that describe a straight line when plotted on a graph. They typically take the form \( ax + by + cz = d \), where \( a \), \( b \), and \( c \) are constants, and \( x \), \( y \), and \( z \) are variables that can change. This form can represent lines and planes in space.

In the context of solving systems of linear equations, you'll usually be dealing with multiple equations at once. Each equation in the set is interconnected, representing different constraints that have to be satisfied simultaneously.

• Each equation represents a unique linear relationship.
• The aim is to find values for the variables that satisfy all equations at once.

Linear equations are fundamental in mathematics, playing a crucial role in various fields such as physics, economics, and engineering.

Substitution Method

The substitution method is a technique used to solve systems of equations, where one equation is solved for one variable in terms of the others. This substitution is then used in the other equations to reduce the number of variables and simplify the system.

Here's a brief rundown of how it works:

• Solve one of the equations for one variable, such as \( y = 1 - 4x \) in equation 4.
• Substitute this expression into another equation, replacing the variable. This creates an equation with one less variable.
• Solve this new equation. Once a variable is found, substitute it back into any previous equation to solve for another variable.

The substitution method is intuitive and helpful when one of the equations is easy to rearrange. It's a powerful tool for solving simultaneous equations.

Elimination Method

The elimination method focuses on removing a variable by adding or subtracting equations. This process simplifies the system of equations, helping to find a solution.

To use the elimination method effectively, follow these steps:

• Align and label your equations to identify matching or opposite terms.
• Adjust equations if necessary, to ensure the elimination of a variable. This often involves multiplying an entire equation by a suitable number.
• Add or subtract equations to eliminate one variable, as seen when eliminating \( z \) from Equations 1 and 2.
• Solve the simplified system, now with fewer variables.

The elimination method is efficient for solving larger systems. By systematically reducing the number of variables, it guides you to the solution of simultaneous equations.

Simultaneous Equations

Simultaneous equations are sets of equations that share the same variables and are solved together. The solutions must satisfy all the equations in the system at the same time.

In real-world scenarios, simultaneous equations help describe systems where multiple conditions or factors interact. For instance, in economic models or engineering constraints.

• The primary goal is to find a common solution that satisfies each equation in the system.
• The solutions are often obtained through methods like substitution and elimination.
• Verification is crucial to ensure that the discovered solution satisfies all the given equations.

By solving simultaneous equations, you can better understand complex relationships and constraints in various fields.