Problem 29 - Algebra and Trigonometry with Analytic Geometry

Question

1-30: Use the method of substitution to solve the system. $$ \left\\{\begin{array}{rr} x+2 y-z= & -1 \\ 2 x-y+z= & 9 \\ x+3 y+3 z= & 6 \end{array}\right. $$

Step-by-Step Solution

Answer

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

Step 1: Solve the First Equation for x

We'll start by solving the first equation for \(x\). The equation is: \[ x + 2y - z = -1 \] Rearrange the equation to express \(x\) explicitly: \[ x = -1 - 2y + z \]

Step 2: Substitute x in the Second Equation

Substitute the expression for \(x\) from Step 1 into the second equation:\[ 2x - y + z = 9 \].Replace \(x\) with \(-1 - 2y + z\): \[ 2(-1 - 2y + z) - y + z = 9 \]Simplify the equation:\[ -2 - 4y + 2z - y + z = 9 \]Combine like terms:\[ -5y + 3z = 11 \]

Step 3: Substitute x in the Third Equation

Substitute the expression for \(x\) from Step 1 into the third equation: \( x + 3y + 3z = 6 \).Replace \(x\) with \(-1 - 2y + z\): \( -1 - 2y + z + 3y + 3z = 6 \)Simplify:\( -1 + y + 4z = 6 \)Rearrange:\( y + 4z = 7 \)

Step 4: Solve the New System of Equations

Now we have a system of equations with two variables:1. \( -5y + 3z = 11 \)2. \( y + 4z = 7 \)Solve the second equation for \(y\): \( y = 7 - 4z \).Substitute \(y = 7 - 4z\) into the first equation:\( -5(7 - 4z) + 3z = 11 \)Simplify and solve for \(z\):\( -35 + 20z + 3z = 11 \)\( 23z = 46 \)\( z = 2 \)

Step 5: Find y using z

Now that we know \(z = 2\), substitute it back into \( y = 7 - 4z \):\( y = 7 - 4(2) \)\( y = 7 - 8 \)\( y = -1 \)

Step 6: Find x using y and z

Use the values \(y = -1\) and \(z = 2\) in the expression found in Step 1: \( x = -1 - 2y + z \)Substitute the values:\( x = -1 - 2(-1) + 2 \)\( x = -1 + 2 + 2 \)\( x = 3 \)

Step 7: Verify the Solution

Verify the solution \(x = 3, y = -1, z = 2\) in the original equations:1. \( x + 2y - z = -1 \) becomes \( 3 + 2(-1) - 2 = -1 \)2. \( 2x - y + z = 9 \) becomes \( 2(3) - (-1) + 2 = 9 \)3. \( x + 3y + 3z = 6 \) becomes \( 3 + 3(-1) + 3(2) = 6 \)Each equation is satisfied, confirming our solution is correct.

Key Concepts

System of Equations

When you're faced with a system of equations, it means you have a set of two or more equations with multiple variables that you need to solve simultaneously. In our exercise, we are dealing with a system of three equations that include the variables \( x \), \( y \), and \( z \). Solving a system of equations means finding the values of these variables that satisfy all the given equations at the same time. This is like finding the intersection point where all the equations "agree" on the values of the variables. Understanding the system is crucial:

• The equations can be linear, which means each term is either a constant or the product of a constant and a single variable.
• In our case, each equation is linear, making it a system of linear equations.

Solving such systems requires specific methods, and for this example, we have used the substitution method.

Solving Linear Equations

Linear equations are mathematical statements of equality that involve linear expressions. The simplicity of these equations lies in the fact that each variable is raised to the first power and their graphs are straight lines. In our exercise, solving linear equations is key to breaking down the larger system into smaller, more manageable parts.Here's how:

• Begin by isolating one of the variables in a simple equation. This is what we did with the first equation to solve for \( x \) in terms of \( y \) and \( z \): \( x = -1 - 2y + z \).
• Once a variable is isolated, replace it in the other equations. This substitution helps reduce the complexity by transforming a system of three equations into a more straightforward problem with fewer variables.

This focused approach leverages the simplicity of linear equations to make the problem more navigable.

Algebraic Techniques

Algebraic techniques allow for strategic manipulation of equations to reveal values of unknowns. Particularly in solving systems of equations, methods like substitution, elimination, and matrix operations come into play. For our example, we specifically applied the substitution method.Substitution Method Explained:

• This technique involves solving one of the equations for a variable and substituting this expression into other equations.
• For instance, after solving the first equation for \( x \), we substituted \( x = -1 - 2y + z \) into the second and third equations. This gives us new equations with just \( y \) and \( z \).
• Reducing the number of variables is key as it simplifies the subsequent steps in solving the system.

These algebraic strategies are powerful tools that help solve complex systems of equations systematically and efficiently.