Problem 56 - Introductory Algebra for College Students

Question

If \(x=3-y-z, 2 x+y-z=-6,\) and $$3 x-y+z=11,\( find the values for \)x, y,\( and \)z$$

Step-by-Step Solution

Answer

The solutions are \(x=3\), \(y=\frac{1}{8}\) and \(z=-\frac{1}{8}\)

Step 1: Substitute the first equation into the second and third equations

You get: - In the second equation: \(2(3-y-z)+y-z=-6\), simplify to get: \(5y+5z=0\), or \(y+z=0\). - In the third equation: \(3(3-y-z)-y+z=11\), simplify to get: \(8y-8z=2\), or \(y-z=\frac{1}{4}\)

Step 2: Solve the two new equations

Solving the two new equations \(y+z=0\) and \(y-z=\frac{1}{4}\), you get: - By adding the two equations, the result is \(2y=\frac{1}{4}\), which simplifies to \(y=\frac{1}{8}\). - By substituting \(y=\frac{1}{8}\) into the equation \(y+z=0\), you get \(z=-\frac{1}{8}\)

Step 3: Substitute \(y\) and \(z\) back into the first equation

Substitute \(y=\frac{1}{8}\) and \(z=-\frac{1}{8}\) back into the first equation \(x=3-y-z\), you get \(x=3-\frac{1}{8}+\frac{1}{8}=3\)

Key Concepts

Simultaneous Equations

Simultaneous equations are a set of equations with multiple variables that we solve together. This means finding a set of values for the variables that satisfy all equations at once. In many cases, simultaneous equations appear in pairs or tuples.
These equations are incredibly useful when we need to model real-world scenarios that involve several unknown quantities. To solve them, we employ methods like substitution or elimination. Each of these methods differs in approach but ultimately serves the goal of finding the values of the variables efficiently.
In our example, we began with three equations containing three variables: \(x\), \(y\), and \(z\). These equations must hold true at the same time for the same values of the variables.
The great thing about solving simultaneous equations is that it not only helps in finding variable values but also enhances our understanding of the relationships between these variables. Through solving, the mutual dependencies across equations become clearer.

Substitution Method

The substitution method is a common technique for solving simultaneous equations. This method involves solving one of the equations for one variable in terms of others and then substituting this expression into the other equations.
It is particularly efficient when an equation is easily manipulable or when variables are already isolated.
In the provided solution, the substitution starts by solving the first equation for \(x\), where \(x = 3 - y - z\). We then substituted this expression for \(x\) into the second and third equations.
This substitution leads to new equations just with \(y\) and \(z\), reducing the complexity of the problem.

• In the second equation: \(2(3-y-z)+y-z=-6\), simplifies to \(y+z=0\).
• In the third equation: \(3(3-y-z)-y+z=11\), simplifies to \(y-z=\frac{1}{4}\).

Using substitution simplifies the system significantly, breaking down multi-variable equations into a more manageable form.

Solve Linear Equations

Solving linear equations means finding the value of the variable that makes the equation true.
This process is foundational in algebra because it allows us to determine unknown quantities using known values and relationships.
In the example provided, we had the equations \(y+z=0\) and \(y-z=\frac{1}{4}\). Solving them required a systematic approach.
By adding these two linear equations, we eliminated \(z\) and isolated \(y\). This gave us \(2y=\frac{1}{4}\), which simplifies to \(y=\frac{1}{8}\).
Next, substituting \(y=\frac{1}{8}\) back into the equation \(y+z=0\), we found \(z=-\frac{1}{8}\). Finally, with these values, we substituted back into the expression for \(x\): \(x=3-\frac{1}{8}+\frac{1}{8}=3\).

Solving linear equations with precision ensures that all calculations correctly align with each equation's logic and arithmetic. Practice will make this process intuitive, providing critical problem-solving skills that are applicable to various mathematical and real-world problems.