Problem 34 - Algebra for College Students

Question

Give a step-by-step description of how you would finc the value of \(x\) in the solution for the system $$ \left(\begin{array}{rl} x+5 y-z & =-9 \\ 2 x-y+z & =11 \\ -3 x-2 y+4 z & =20 \end{array}\right) $$

Step-by-Step Solution

Answer

The value of \(x\) is 4.

Step 1: Select Two Equations to Eliminate z

Choose the first two equations to eliminate the variable \(z\). We will use the first equation \(x + 5y - z = -9\) and the second equation \(2x - y + z = 11\). By adding these two equations together, the \(z\) terms will cancel out.

Step 2: Add the First Two Equations

Add the first two equations to eliminate \(z\): \[(x + 5y - z) + (2x - y + z) = -9 + 11\] Simplifying gives: \[3x + 4y = 2\]. We have our first equation in \(x\) and \(y\).

Step 3: Eliminate z Using Another Pair of Equations

Choose the second equation \(2x - y + z = 11\) and the third equation \(-3x - 2y + 4z = 20\). We will eliminate \(z\) by making the coefficients of \(z\) equal in magnitude but opposite in sign.

Step 4: Multiply Equations to Equate Coefficients

Multiply the second equation by 4 and the third equation by 1: \[(4)(2x - y + z) = 4(11)\] \[-3x - 2y + 4z = 20\]. Simplifying gives us: \[8x - 4y + 4z = 44\] \[-3x - 2y + 4z = 20\].

Step 5: Subtract to Eliminate z

Subtract the modified third equation from the modified second equation to eliminate \(z\): \[(8x - 4y + 4z) - (-3x - 2y + 4z) = 44 - 20\]. Simplify this to obtain \[11x - 2y = 24\].

Step 6: Solve the System of Two Equations

We now have two equations: \[3x + 4y = 2\] and \[11x - 2y = 24\]. We will solve this system of equations using substitution or elimination method. Let's use the elimination method to solve for \(x\).

Step 7: Eliminate y to Solve for x

Multiply the first equation by 2 to align the coefficients of \(y\): \[2(3x + 4y) = 2(2)\] which simplifies to \[6x + 8y = 4\]. Now subtract the second equation \(11x - 2y = 24\): \[(6x + 8y) - (11x - 2y) = 4 - 24\]. Simplify to find \[-5x + 10y = -20\].

Step 8: Simplify and Solve for x

Rearrange the simplified equation \(-5x = -20\) to \(5x = 20\) and solve for \(x\) by dividing both sides by 5: \[x = 4\].

Key Concepts

Elimination Method

The elimination method is a systematic way to solve systems of linear equations. It involves strategically modifying and combining equations to cancel out one of the variables, making it easier to solve the remaining equations. In our example, we have three equations. To use the elimination method:

• Identify a variable to eliminate. It's often easiest to start with one that’s easily cancelled, like in our case, we chose to eliminate z.
• Add, subtract, or multiply the equations as needed to eliminate one variable. This will give you a reduced system of equations.
• Repeat this process if necessary to eliminate another variable, allowing you to solve for the remaining unknowns.

The elimination method provides a clear pathway to simplify complex systems by reducing them to a more manageable form.

Simultaneous Equations

Simultaneous equations are a set of equations with multiple variables that are solved together, meaning the solution must satisfy all equations at once. For example, the system in our exercise involves three simultaneous equations:

• \(x + 5y - z = -9\)
• \(2x - y + z = 11\)
• \(-3x - 2y + 4z = 20\)

These equations need to be solved such that a single set of values for \(x, y,\) and \(z\) will satisfy all three. Solving simultaneous equations is integral to finding interconnected solutions, which is crucial in many fields including physics and engineering. Solutions often require strategic use of either the elimination or substitution method.

Matrix Representation

Matrix representation is a powerful tool in solving systems of linear equations because it organizes equations into a compact and manageable form. The given system of equations can be expressed as a matrix:
\[\begin{bmatrix}1 & 5 & -1 \2 & -1 & 1 \-3 & -2 & 4\end{bmatrix}\begin{bmatrix}x \y \z\end{bmatrix}= \begin{bmatrix}-9 \11 \20\end{bmatrix}\]Matrices simplify operations through methods like row reduction, allowing for systematic elimination of variables. This approach is especially useful for computer algorithms in solving large systems of equations. Understanding matrix representation helps in visualizing operations and is foundational in linear algebra.

Substitution Method

The substitution method is an alternative approach to solving systems of equations by expressing one variable in terms of others and substituting that into another equation. Here is how it's done:

• Solve one of the equations for one of its variables, which will express this variable as a function of the others.
• Substitute this expression in place of the variable in other equations, effectively reducing the number of variables.
• This process can be repeated until you find the value of one unknown, which can then be used to find others.

Although the substitution method can be more straightforward for smaller systems, it sometimes leads to complex algebra in larger systems. It parallels the elimination method in aiming to reduce the system gradually but through a different mechanism. It’s an excellent tool for systems where one of the equations is easily solvable for a variable.