Problem 4 - Algebra A Combined Function

Question

Solve each system. $$ \left\\{\begin{array}{ll} 5 x & =5 \\ 2 x+y & =4 \\ 3 x+y-4 z & =-15 \end{array}\right. $$

Step-by-Step Solution

Answer

(1, 2, 5)

Step 1: Solve for x from the first equation

The first equation of the system is \(5x = 5\). To solve for \(x\), divide both sides of the equation by 5. This gives:\[x = \frac{5}{5} = 1\]Thus, \(x = 1\).

Step 2: Substitute x in the second equation to solve for y

Substitute \(x = 1\) into the second equation, \(2x + y = 4\):\[2(1) + y = 4\]This simplifies to:\[2 + y = 4\]Subtract 2 from both sides to solve for \(y\):\[y = 4 - 2 = 2\]Thus, \(y = 2\).

Step 3: Substitute x and y into the third equation to solve for z

Substitute \(x = 1\) and \(y = 2\) into the third equation, \(3x + y - 4z = -15\):\[3(1) + 2 - 4z = -15\]Simplify the equation:\[3 + 2 - 4z = -15\]Combine like terms:\[5 - 4z = -15\]Subtract 5 from both sides:\[-4z = -15 - 5\]Simplify the right side:\[-4z = -20\]Divide both sides by -4 to solve for \(z\):\[z = \frac{-20}{-4} = 5\]Thus, \(z = 5\).

Step 4: Write the solution as an ordered triplet

The solution to the system of equations is the values of \(x\), \(y\), and \(z\). Thus, the solution is:\((x, y, z) = (1, 2, 5)\).

Key Concepts

Linear Equations

A linear equation is a mathematical statement that describes a straight line when graphed on a coordinate plane. It is expressed in the form of an equation where each term is either a constant or the product of a constant and a single variable. The general format for a single-variable linear equation is: \( ax + b = 0 \), where \( a \) and \( b \) are constants.
Linear equations can include one or more variables, and when there are multiple equations working together, they form a system of equations. For example, in the system given in the exercise, we have three equations:

• \( 5x = 5 \)
• \( 2x + y = 4 \)
• \( 3x + y - 4z = -15 \)

All these equations are linear equations because they involve the first power of variables \( x \), \( y \), and \( z \), without any exponents or nonlinear combinations. Understanding how to work with linear equations is fundamental when solving for unknown variables.

Substitution Method

The substitution method is a technique used to solve systems of equations, whereby one equation is solved for one variable in terms of the others and then substituted into the other equations. This method can simplify the process of finding the common solution that satisfies all equations in the system.
In the original exercise, we first solved the equation \( 5x = 5 \) to find \( x = 1 \). This solution for \( x \) was then substituted into the second equation \( 2x + y = 4 \), which helped us find \( y = 2 \). By repeating this process, substituting the values of \( x \) and \( y \) into the third equation \( 3x + y - 4z = -15 \), we determined that \( z = 5 \).

• This approach allows each variable to be expressed explicitly, making the equations progressively simpler at each step.
• It helps to systematically reduce the system to a simpler form, often down to a single equation with one variable.

The substitution method is particularly helpful when one of the equations is easily solvable for one of the variables, as we did for \( x \) in this exercise.

Ordered Triplet

An ordered triplet is a set of values that correspond to the solution of a system of three equations involving three variables. These values represent the simultaneous solution for each variable in the system of equations. In our case, the ordered triplet is \((x, y, z) = (1, 2, 5)\).
This notation is significant because:

• It indicates the specific values of \( x \), \( y \), and \( z \) that satisfy all of the equations together.
• It clearly communicates the relationship between the variables once the solution is attained.

By finding this ordered triplet, we confirm that our solution is correct if substituting these values back into each of the original equations results in true statements.
Ordered triplets are quite useful not just in linear algebra but also in real-world contexts such as in systems modeling, engineering, economics, and anywhere multiple linear relationships need to be analyzed together. Understanding and using ordered triplets effectively can enhance your capability to handle complex mathematical challenges.