Problem 14 - Algebra and Trigonometry

Question

Solve each system. $$ \left\\{\begin{aligned} x+3 y+5 z &=20 \\ y-4 z &=-16 \\ 3 x-2 y+9 z &=36 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The solution to the system of equations is \( x = 44 \), \( y = 0 \), and \( z = 4 \)

Step 1: Express y from Equation 2

First, isolate y from Equation 2 to make it easier to substitute y into other equations. This can be done by rearranging equation 2 as \( y = 4z -16 \).

Step 2: Substitute y into Equation 1 and 3

Substitute \( y = 4z - 16 \) into Equation 1 and 3, which gives us:From equation 1, we get: \( x+3(4z -16)+5z = 20 \), simplify to get \( x = -7z + 68 \).From equation 3, we get: \( 3x - 2(4z - 16) + 9z = 36 \), substitute \( x = -7z + 68 \) in to get \( 3(-7z + 68) - 8z + 32 = 36 \), solve to get \( z = 4 \).

Step 3: Backward Substitution

Now substitute \( z = 4 \) into the isolations of \( x \) and \( y \) from step 2, to find the values of \( x \) and \( y \). Therefore, \( x = -7*4 + 68 = 44 \) and \( y = 4*4 - 16 = 0 \).

Step 4: Conclusion

So, the solution to the system of equations is \( x = 44 \), \( y = 0 \), and \( z = 4 \)

Key Concepts

Linear Equations

Linear equations are mathematical expressions involving variables raised to the power of one. They form a straight line when graphed. In a system of linear equations, we deal with multiple such equations that work together to determine a common solution for all included variables.
In our specific system of equations:

• Equation 1: \( x + 3y + 5z = 20 \)
• Equation 2: \( y - 4z = -16 \)
• Equation 3: \( 3x - 2y + 9z = 36 \)

Each equation contains three variables: \( x \), \( y \), and \( z \). The key is to find the values of these variables that satisfy all equations simultaneously.

Substitution Method

The substitution method is a powerful technique for solving systems of equations. It involves solving one of the equations for one variable and then substituting this expression into the other equations.
This method is useful because it reduces the number of variables, making the system easier to solve. Here, we solved Equation 2 for \( y \):

• \( y = 4z - 16 \).

Then, this expression for \( y \) was substituted into Equations 1 and 3, allowing us to express \( x \) in terms of \( z \) and eventually solving for \( z \).

Solution of Equations

Finding the solution of the system involves determining the specific values for the variables that satisfy each linear equation.
Substituting \( y = 4z - 16 \) into Equation 1 gives:

• \( x + 3(4z - 16) + 5z = 20 \)
• Simplifying gives: \( x = -7z + 68 \)

Substituting into Equation 3 gives:

• \( 3x - 2(4z - 16) + 9z = 36 \)
• With x substituted, solving reveals \( z = 4 \)

This process progressively narrows down the values of \( x \), \( y \), and \( z \).

Backward Substitution

Backward substitution is the final step where you plug the known value of one variable back into the equations set aside from previous steps to find the other unknowns.
After finding \( z = 4 \), substitute back:

• \( x = -7 * 4 + 68 \) gives \( x = 44 \)
• \( y = 4 * 4 - 16 \) gives \( y = 0 \)

In conclusion, using this systematic approach ensures we find precise solutions, confirming \( x = 44 \), \( y = 0 \), and \( z = 4 \). Each value checks out against the original system of equations, proving the solution's validity.