Problem 56 - Algebra A Combined Function

Question

Solving systems involving more than three variables can be accomplished with methods similar to those encountered in this section. Apply what you already know to solve each system of equations in four variables. $$ \left\\{\begin{aligned} 2 x \quad-z &=-1 \\ y+z+w &=9 \\ y-2 w &=-6 \\ x+y &=3 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = 4\), \(y = -1\), \(z = 9\), and \(w = 2\).

Step 1: Express Variables in Terms of Others

From the first equation, express \(x\) in terms of \(z\): \[ x = \frac{z-1}{2} \] From the last equation, express \(y\) in terms of \(x\):\[ y = 3 - x \] Substitute \(x\) from the previous expression: \[ y = 3 - \frac{z - 1}{2} = \frac{5}{2} - \frac{z}{2} \]

Step 2: Substitute to Reduce Variables

Substitute \(y\) from the Step 1 into equations 2 and 3:Equation 2:\[ \frac{5}{2} - \frac{z}{2} + z + w = 9 \] Equation 3:\[ \frac{5}{2} - \frac{z}{2} - 2w = -6 \]

Step 3: Simplify Equations

Simplify Equation 2:\[ \frac{5}{2} + \frac{z}{2} + w = 9 \]Multiply everything by 2:\[ 5 + z + 2w = 18 \]Rearrange:\[ z + 2w = 13 \] Simplify Equation 3:\[ \frac{5}{2} - \frac{z}{2} - 2w = -6 \]Multiply everything by 2:\[ 5 - z - 4w = -12 \]Rearrange:\[ z + 4w = 17 \]

Step 4: Solve the Simplified System

Now, solve the system of equations: \( z + 2w = 13 \) \( z + 4w = 17 \) Subtract the first from the second:\[ (z + 4w) - (z + 2w) = 17 - 13 \]\[ 2w = 4 \]\[ w = 2 \]Substitute \(w\) back into one of the equations to solve for \(z\):\[ z + 2(2) = 13 \]\[ z + 4 = 13 \]\[ z = 9 \]

Step 5: Solve for Remaining Variables

Now use \(z = 9\) in the earlier found expression for \(x\):\[ x = \frac{9 - 1}{2} = 4 \] Finally, use \(x = 4\) in the expression for \(y\):\[ y = 3 - 4 = -1 \]

Key Concepts

Solving Systems with Multiple Variables

Breaking down a system of equations with several variables is a step-by-step adventure. Start by recognizing how many variables you are working with and then arrange the given equations in a way that speaks to these variables. Our goal is to gradually simplify the problem. You do this by expressing some variables in terms of others. This manipulation will make it possible to reduce the overall number of variables. Once you're down to two variables or even fewer, you can solve the system more easily.
As we transition through each step, it's crucial to choose which variable to eliminate strategically. You can repeatedly pick off one variable at a time until all are determined. The magic happens when we use substitution effectively to reduce unwanted complexity in the system.

Step-by-Step Solutions

Understanding how to tackle systems of equations through step-by-step solutions is like following a recipe. Each step provides a vital clue or action that helps to isolate the variables. Begin by solving one of the equations for a chosen variable. Experience shows that this makes other equations more manageable.
In the given exercise, the solution starts with simplifying one equation to express a variable like \( x \) in terms of \( z \). Once this is established, substitute it back into another equation to reduce it further. Keep applying this method and substitute the simplified terms into the remaining equations. This will systematically decrease the number of variables in the system.
After reducing the complexity, you often reach equations with fewer variables. Solve them like a series of simpler equations, keeping track of each solution, ensuring that you go back to substitute fully until every original variable has a known value.

Algebraic Expressions

Algebraic expressions provide the language that allows us to express systems of equations in concise, useful formats. They show relationships between variables in symbolic form and can be manipulated to reveal more about the solutions to the equations you are dealing with.
Take the time to re-arrange initial algebraic expressions in the system. For example, expressions like \( x = \frac{z - 1}{2} \) can be extremely useful for making substitutions that simplify both the equations and the problem. Remember, algebraic operations like addition, subtraction, and multiplication help in forming new expressions to substitute into equations, breaking down larger problems into solvable parts.
Combining algebraic operations and expressions allows you to rewrite and solve many practical problems. In our case, we simplify using multiplication by constants to clarify relationships and solve multi-variable systems efficiently. Having a knack for playing with algebraic expressions is key to cracking complicated systems of equations.