Problem 49 - Algebra Form and Function

Question

Solve the system of equations $$ \left\\{\begin{aligned} 3 x+2 y+5 z &=11 \\ 2 x-3 y+z &=7 \\ z &=2 x . \end{aligned}\right. $$ Hint: Use the third equation to substitute for \(z\) in the other two.

Step-by-Step Solution

Answer

Question: Find the values of x, y, and z that satisfy the following system of equations: $$ \left\\{\begin{aligned} 3x + 2y + 5z &= 11, \\\ 2x - 3y + z &= 7, \\\ z &= 2x. \end{aligned}\right. $$ Answer: The solution for the system of equations is x = 1, y = -1, and z = 2.

Step 1: Substitute the third equation in the other two equations

Replace \(z\) with \(2x\) in the first two equations using the third equation, obtaining: $$ \left\\{\begin{aligned} 3x + 2y + 5(2x) &= 11, \\\ 2x - 3y + 1(2x) &= 7. \end{aligned}\right. $$

Step 2: Simplify the new system of equations

Now, simplify the two equations by expanding and re-arranging terms: $$ \left\\{\begin{aligned} 3x + 2y + 10x &= 11 \Rightarrow 13x + 2y &= 11, \\\ 2x - 3y + 2x &= 7 \Rightarrow 4x - 3y &= 7. \end{aligned}\right. $$

Step 3: Solve the system of two linear equations with two variables

Using the substitution or elimination method, we can solve the system of equations: 1. Multiply the first equation by 3 so that the coefficients of \(y\) in both equations become the same: $$ (13x + 2y)*3 = 11*3 \\ 39x + 6y = 33. $$ 2. Now, we have the following system: $$ \left\\{\begin{aligned} 39x + 6y &= 33, \\\ 4x - 3y &= 7. \end{aligned}\right. $$ 3. Add the second equation multiplied by 2 to the first equation to eliminate \(y\) and find \(x\): $$ 2(4x - 3y) + (39x + 6y) = 2(7) + 33 \\ 8x - 6y + 39x + 6y = 14 + 33 \\ 47x = 47. $$ 4. Divide both sides by 47 to get the value of \(x\): $$ x = \frac{47}{47} = 1. $$

Step 4: Find the value of \(y\)

Now, substitute the value of \(x\) in the second equation to find the value of \(y\): $$ 4(1) - 3y = 7 \\ 4 - 3y = 7 \\ -3y = 3 \\ y = \frac{3}{-3} = -1. $$

Step 5: Find the value of \(z\)

Finally, substitute the value of \(x\) in the third equation to find the value of \(z\): $$ z = 2(1) \\ z = 2. $$ So, the solution for the system of equations is \(x = 1\), \(y = -1\), and \(z = 2\).

Key Concepts

Substitution Method

The substitution method is a fundamental technique for solving a system of equations.
It involves solving one of the equations for one variable and then substituting this expression into the other equations. This systematically reduces the number of equations and variables, simplifying the process.
In the original problem, the third equation is the most straightforward: \(z = 2x\).
By solving for \(z\) and substituting \(z = 2x\) into the other two equations, we transform the system into one with just two variables, \(x\) and \(y\).
This helps to create a simplified system of linear equations.Steps to use the substitution method:

• Identify the simplest equation to solve for one variable.
• Solve this equation for the chosen variable.
• Substitute this expression into the other equations.
• Simplify the resulting equations and solve for the remaining variables.

Simplifying a system of equations makes them easier to handle and solve.

Elimination Method

The elimination method is another powerful tool for solving systems of linear equations.
This strategy involves adding or subtracting equations to eliminate one of the variables. This makes it increasingly straightforward to determine solutions for the remaining variables.
In our problem, once the substitution method has reduced the system to two equations with two variables, the elimination method can be effectively used.
By multiplying the first equation by 3, the coefficients of \(y\) become equal and thus, the equations can be combined to eliminate \(y\) from the system.Steps to use the elimination method:

• Multiply one or both equations, if necessary, to obtain the same coefficient of one variable.
• Add or subtract the equations to eliminate one variable.
• Once a variable is eliminated, solve for the remaining variable.
• Back-substitute to find other variable values.

This method is especially useful when the system reduces to a smaller number of variables.

Linear Equations

Linear equations are mathematical expressions that represent straight lines when graphed on a coordinate plane.
They have the general form \(ax + by + cz = d\), where \(a\), \(b\), \(c\), and \(d\) are constants, and \(x\), \(y\), and \(z\) are variables.
Solving a system of linear equations involves finding the values of the variables that satisfy all equations simultaneously.
Linear equations often interact at a specific point, which represents the solution of the system when graphed.In the given problem, we have three linear equations featuring three variables \(x\), \(y\), and \(z\).
It's crucial to manipulate these equations using methods like substitution and elimination to find the correct values for the variables.
Characteristics of linear equations include:

• They can be solved using basic algebraic techniques.
• They represent straight lines in a graph.
• They can be used to describe a wide variety of real-world scenarios.

Understanding these aspects makes solving them much clearer and manageable.