Problem 55 - 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. $$ \left\\{\begin{aligned} x+y+z+w &=5 \\ 2 x+y+z+w &=6 \\ x+y+z &=2 \\ x+y &=0 \end{aligned}\right. $$

Step-by-Step Solution

Answer

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

Step 1: Analyze the System

Examine the given system of equations to look for easy substitutions or simplifications. Notice that the fourth equation is \(x + y = 0\). This can be quickly solved to express one variable in terms of another. Let's solve the equation \(x + y = 0\) for \(y\).

Step 2: Substitute to Reduce Variables

From \(x + y = 0\), we have \(y = -x\). Substitute \(y = -x\) into the other equations. For the third equation \(x + y + z = 2\), it becomes \(x - x + z = 2\) or \(z = 2\). Now substitute for \(y\) and \(z\) in the remaining two equations.

Step 3: Substitute in the First Two Equations

Substitute \(y = -x\) and \(z = 2\) into the first equation. \(x + (-x) + 2 + w = 5\) simplifies to \(2 + w = 5\), hence \(w = 3\). Then substitute the same values into the second equation: \(2x + (-x) + 2 + w = 6\) simplifies to \(x + 2 + w = 6\), substitute \(w = 3\) to get \(x + 2 + 3 = 6\).

Step 4: Solve for Remaining Variables

Continue solving the simplified equation \(x + 2 + 3 = 6\), giving \(x = 1\). Now use this value to find the others: since \(y = -x\), \(y = -1\), and \(z\) was calculated as 2, \(w = 3\). Thus, the solution is: \(x = 1\), \(y = -1\), \(z = 2\), and \(w = 3\).

Key Concepts

Four Variables

In algebra, solving systems of equations involving four variables is a bit more challenging, as they require balancing more equations and maintaining more variables at the same time. Imagine trying to juggle four different balls in the air instead of just two or three.
Each variable, often represented by the symbols like \(x\), \(y\), \(z\), and \(w\), means we are trying to find four different unknown values that will satisfy all the given equations at once. This is like looking for a single key that can unlock four locks simultaneously.
To do this successfully, you must identify relationships or dependencies between the variables to solve them step by step, gradually narrowing down to find each variable's actual value. With practice, this process becomes more intuitive, even with a higher number of variables.

Substitution Method

The substitution method is a key strategy in algebra when tackling systems of equations. It involves replacing one variable with an equivalent expression containing another variable.
This method makes it possible to reduce the number of variables step by step.
In our exercise, this strategy starts with the equation \(x + y = 0\). From here, we solve for \(y\) in terms of \(x\), which leads us to: \(y = -x\). Once we have this, we can substitute \(y\) in the other equations. This process simplifies the system by focusing only on a few variables at a time.
If you think of each substitution step as peeling off a layer of an onion, it allows you to zero in on the core solution more easily. It's a method that may seem complex at first, but it makes solving multi-variable systems much more manageable with practice.

Algebra

Algebra serves as the backbone for solving complicated systems like the one we're dealing with. It allows us to manipulate equations, transform them, and ultimately find solutions for the variables.
With a toolbox full of operations such as addition, subtraction, multiplication, and division, you apply each operation methodically to simplify equations and solve for unknowns.
This foundational knowledge in algebra enables you to tackle broader mathematical problems and develop a logical, step-by-step approach to problem-solving. Algebra is essentially about finding patterns and solutions in diverse situations. Whether you're working through equations with fewer variables or progressing to those with four or more, algebra gives you the framework to tackle all these problems with consistency and accuracy.

Step by Step Solution

Understanding a step-by-step solution is crucial when solving more complex equations. It allows you to break down a challenging system into smaller, more manageable parts.
Here's how you can visualize the process:

• **Step 1: Analyze the System.** Look for any equations that can quickly reduce one or more variables.
• **Step 2: Substitute to Reduce Variables.** Use known equations to express variables as functions of others.
• **Step 3: Substitute in Remaining Equations.** Reduce complexity by substituting simplified expressions into other equations.
• **Step 4: Solve for Remaining Variables.** Continuously substitute and solve until all variables are known.

This guide helps cement your understanding, transforming a seemingly insurmountable problem into a series of achievable steps. Remember, patience and precision are key—step-by-step solutions teach us to approach intricate tasks with care and logic.