Problem 38 - College Algebra Essentials

Question

Solve the system: $$\begin{aligned} w-x+2 y-2 z &=-1 \\ x-\frac{1}{3} y+z &=\frac{8}{3} \\ y-z &=1 \\ z &=3 \end{aligned}$$ Express the solution set in the form \(\\{(w, x, y, z)\\} .\) What makes it fairly easy to find the solution?

Step-by-Step Solution

Answer

The solution to the system of equations is: \{(2, 1, 4, 3)\} . The presence of a directly known variable (z) makes it fairly easy to find the solution, as it can be substituted into the other equations, simplifying them considerably and allowing for sequential solving.

Step 1: Substituting value of z

Given that \(z = 3\), substitute this into the other equations: \[\begin{aligned} w-x+2 y-2 z &=-1 \ x-\frac{1}{3} y+z &=\frac{8}{3} \ y-z&=1 \end{aligned}\] becomes \[\begin{aligned} w-x+2 y-6 &=-1 \ x-\frac{1}{3} y+3 &=\frac{8}{3} \ y-3&=1 \end{aligned}\] respectively.

Step 2: Solving for y

From the third equation, solve for \(y\). This gives \(y = 1 + 3 = 4\).

Step 3: Substituting value of y into second equation and to solve for x

Now, substitute \(y = 4\) into the second equation, \(x-\frac{1}{3} y+3 =\frac{8}{3}\) to solve for \(x\). This gives \(x = 1\).

Step 4: Substituting values of x, y and z into the first equation to solve for w

Finally, substitute \(x = 1\), \(y = 4\) and \(z = 3\) into the first equation, to solve for \(w\). This gives \(w = 2\).

Key Concepts

Algebraic Substitution

Understanding algebraic substitution is crucial when you're working with systems of equations. It's a method where you use the known values of some variables to find the unknown values of others. In the exercise provided, we see this method in action.

Once the value of one variable is known—in this case, the equation thirdly informs us that \( z = 3 \)—we substitute this value into the other equations. It's like having a key that unlocks parts of the puzzle one by one. Algebraic substitution not only simplifies the other equations but also guides us closer to the entire solution set.

Why Substitute?

• Simplification: Replacing variables with known values make the equations less complex.
• Clarity: It can reduce the system to fewer equations with fewer variables, making it easier to see the next steps.
• Systematic Approach: Substitution provides a clear pathway through the system, finding one variable’s value at a time.

Using algebraic substitution smartly, as in stepping down from four variables to one, can make an intimidating system of equations feel much more manageable.

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. Here, we're looking at a system involving four variables: \( w, x, y, \) and \( z \). Systems like these can have one unique solution, no solution, or infinitely many solutions.

In the context of the provided exercise, our goal was to find unique values for each variable that satisfy all four equations simultaneously. We achieved this by systematically solving the given linear equations using substitution. Such systems are foundational in algebra—they model many real-world scenarios where multiple conditions must be met simultaneously.

Elementary Algebra

Elementary algebra is the backbone of understanding how to manipulate equations to solve for unknowns. In our example, we used elementary algebra techniques to isolate variables and solve equations sequentially.

Starting with basic arithmetic and moving through operations like addition, subtraction, multiplication, division, and understanding equalities and inequalities, foundational algebra equips us with the tools to handle systems of equations. It's about recognizing patterns and applying the laws of arithmetic consistently to both sides of an equation, ensuring that whatever changes you make, the equation remains balanced.

Simultaneous Equations Solution

Arriving at a simultaneous equations solution is the ultimate goal when we're faced with a set of equations that depend on each other. This solution is a set of values that satisfies all the equations at once.

For the provided exercise, when we plug the values back into the original equations and they hold true, we know we've found the correct solution. The solution set in the form \( \{ (w, x, y, z) \} \) represents the specific values that all four variables must have to satisfy the given equations. The exercise simplifies this process by giving us the value for one variable outright, highlighting that the nature of how information is presented in a problem can greatly ease the task of finding a simultaneous solution.