Problem 27 - Precalculus with Limits

Question

Solving a System of Linear Equations In Exercises \(25 - 46\) , solve the system of linear equations and check any solutions algebraically. $$\left\\{ \begin{aligned} 2 x & \+ 2 z = 2 \\ 5 x + 3 y & = 4 \\ 3 y - 4 z & = 4 \end{aligned} \right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 9/4\), \(y = -1/3\), and \(z = -5/4\).

Step 1: Simplify The First Equation

Notice that the first equation, \(2x + 2z = 2\), can be simplified by dividing through by 2 to give: \(x + z = 1\).

Step 2: Substitute First Equation Into Second Equation

Take the simplified first equation and substitute it into the second equation. This will give: \(5(x + z) + 3y = 4\). This simplifies to \(5 + 3y = 4\), which can further be simplified to give \(3y = -1\). Solve this for \(y\) to give \(y = -1/3\).

Step 3: Substitute \(y\) Into Third Equation

Substitute \(y = -1/3\) into the third equation to get \(3(-1/3) -4z = 4\). Simplify this equation to give \(-4z = 5\) which further simplifies to \(z = -5/4\).

Step 4: Substitute \(z\) Into First Equation

Substitute \(z = -5/4\) into the first simplified equation \(x + z = 1\). Simplify to get \(x = 1 -(-5/4)\), which simplifies to \(x = 9/4\).

Step 5: Verify Solution

Verify the solution by substituting \(x = 9/4\), \(y = -1/3\), and \(z = -5/4\) into the original system of equations. If the values of \(x\), \(y\), and \(z\) satisfy all equations in the original system, the solution is verified.

Key Concepts

Algebraic Substitution Method

The algebraic substitution method is a powerful technique used to solve systems of linear equations. When dealing with multiple variables, finding a solution can seem daunting. However, substitution allows us to reduce the system to one with fewer variables by expressing one variable in terms of another.

Here's the process illustrated: Imagine you have the simplified equation of a system, such as \(x + z = 1\). You can express \(z\) as \(z = 1 - x\) and substitute this expression into another equation in the system. For example, if the second equation is \(5x + 3y = 4\), replacing \(z\) with \(1 - x\) would result in an equation with only \(x\) and \(y\). This method reduces the complexity of the problem and eventually allows you to solve for all variables, one by one.

Simplifying Equations

Simplifying equations is an essential step in solving algebraic problems. It involves reducing the complexity of the equation to its most basic form without changing its solution. For instance, take the equation \(2x + 2z = 2\). This can be simplified by dividing each term by 2, which gives \(x + z = 1\).

Getting into this habit does more than just make the numbers smaller and easier to work with; it also helps to identify the structure of the equation, making substitution clearer. Simplification often includes combining like terms, factoring, and reducing fractions. The goal is to make the next steps of solving the equation, whether that's substitution or another method, as straightforward as possible.

Verifying Solutions Algebraically

Once you believe you have found the solution to a system of equations, it's crucial to verify that the solution is correct. Verifying solutions algebraically involves substituting the values you have found back into the original equations to check if they satisfy every equation in the system.

For example, if you have found solutions \(x = 9/4\), \(y = -1/3\), and \(z = -5/4\), you would replace each variable in the original equations with these values. If each equation balances out, your solution is correct. It's a good habit to always perform this step before considering a problem finished, as it ensures that no errors were made during the problem-solving process.