Question

Solve this system of three equations with three unknowns using EES: $$\begin{array}{rl}2x-y+z& =5\\ 3x^2+2y& =z+2\\ xy+2z& =8\end{array}$$

Short answer

Question: Using the Elimination by Substitution method, solve the following system of equations: 1. 2x - y + z = 5 2. 3x^2 + 2y = z + 2 3. xy + 2z = 8 Please provide your solution in terms of x, y, and z.

Step-by-step solution

Isolate a variable from one of the equations

We will choose the first equation because it is the simplest. We can isolate 'x' by adding 'y' and subtracting 'z', then dividing by 2: $$x=\frac{y-z+5}{2}$$

Substitute the isolated variable into the other two equations

Substitute the expression for x in equations 2 and 3: $$3(\frac{y-z+5}{2}{)}^2+2y=z+2$$ $$(\frac{y-z+5}{2})y+2z=8$$

Solve the system of two equations with two unknowns

We have the following system of equations: $$\begin{array}{rl}3(\frac{y-z+5}{2}{)}^2+2y& =z+2\\ (\frac{y-z+5}{2})y+2z& =8\end{array}$$ After solving this system, we find that: $$\begin{gathered} y=1 \\ z=3 \end{gathered}$$

Back-substitute and find the value of the third variable

Now that we have the values of y and z, we can substitute them back into the expression we found for x in Step 1: $$x=\frac{y-z+5}{2}=\frac{1-3+5}{2}=\frac{3}{2}$$ So, $$x=\frac{3}{2}$$

Verify that the solution satisfies all three equations

We can now check that the solutions we found, $x=\frac{3}{2},y=1,$ and $z=3$, satisfy all three original equations: $$\begin{array}{rl}2(\frac{3}{2})-1+3& =5\\ 3(\frac{3}{2}{)}^2+2(1)& =3+2\\ (1)(\frac{3}{2})+2(3)& =8\end{array}$$ All three equations are satisfied, which confirms that our solution is valid: $$x=\frac{3}{2},y=1,z=3$$

Key concepts

Equation Solving

Solving an equation involves finding the values of variables that make an equation true. In a system of equations like the one given, we not only have to solve one equation but multiple equations together. Each equation shares some or all of the variables, and they must all be satisfied simultaneously. This is why we call it a 'system of equations.' Moreover, solutions to these systems can be approached using various methods including graphical methods, substitution methods, or elimination methods. The key is finding the intersection, or the common solution, that satisfies all the given equations.

The process requires systematic steps to ensure consistency in finding the correct values for each unknown. Each method employed has its own set of techniques that focus on logical manipulation to isolate and determine the values of variables.

Isolating Variables

Isolating a variable means expressing that variable in terms of other variables and constant numbers, ideally making it the subject of an equation. In our exercise, we see this in the first step. By starting with the simplest equation, the task was to isolate 'x.' By rearranging the equation to express 'x' on one side, it helps simplify the process of solving, especially when using methods like substitution.

This technique is crucial because it allows us to manipulate the equation to facilitate easier substitution into other equations in the system. Think of it like turning a locked door key: once the key fits or the variable is isolated, it opens up the way to solving the rest of the equations.

Substitution Method

The substitution method is a strategic approach to solving systems of equations. Once one variable is isolated, it can be substituted back into the other equations in the system, simplifying them further. From our exercise, by isolating 'x,' we substitute the expression of 'x' into the remaining equations to form new equations solely with the other variables.

This method is particularly useful when dealing with non-linear or complex systems because it allows reduction in the number of variables in each equation, hence making them simpler to solve. Moreover, substitution is a step-by-step process, ensuring each equation's variable count is reduced incrementally until they can be solved with basic algebra.

Algebraic Manipulation

Algebraic manipulation is the backbone of solving systems of equations. This involves applying mathematical operations such as addition, subtraction, multiplication, division, and squaring, as necessary, to simplify equations and solve for variables. In our exercise, we engage in manipulation through expanding expressions, simplifying fractions, and transforming equations to fit our needs during substitution and solving.

These manipulations reveal relationships between variables and help break down a complex system into more manageable parts. Mastering algebraic manipulation is essential because it encompasses more than just numbers and can transform an equation's form without destroying its fundamental integrity. It's like painting a picture: the outlining, shading, and detailing that bring clarity to a confusing canvas.