Question

Verify that the values of the variables listed are solutions of the system of equations. $$ \begin{array}{l} \left\\{\begin{array}{l} 3 x+3 y+2 z=4 \\ x-y-z=0 \\ 2 y-3 z=-8 \\ \end{array}\right.\\\ x=1, y=-1, z=2 \\ (1,-1,2) \end{array} $$

Short answer

(1, -1, 2) is a solution to the system of equations

Step-by-step solution

Substitute values into the first equation

The first equation is \(3x + 3y + 2z = 4\). Substitute \(x = 1\), \(y = -1\), and \(z = 2\) into this equation.the equation becomes: \(3(1) + 3(-1) + 2(2) = 4\)

Simplify the first equation

Simplify the equation to determine if it holds true.\(3(1) + 3(-1) + 2(2) = 3 - 3 + 4 = 4\).Since the left-hand side equals the right-hand side, the first equation holds true.

Substitute values into the second equation

The second equation is \(x - y - z = 0\). Substitute \(x = 1\), \(y = -1\), and \(z = 2\) into this equation.the equation becomes: \(1 - (-1) - 2 = 0\)

Simplify the second equation

Simplify the equation to determine if it holds true.\(1 - (-1) - 2 = 1 + 1 - 2 = 0\).Since the left-hand side equals the right-hand side, the second equation holds true.

Substitute values into the third equation

The third equation is \(2y - 3z = -8\). Substitute \(y = -1\) and \(z = 2\) into this equation.the equation becomes: \(2(-1) - 3(2) = -8\)

Simplify the third equation

Simplify the equation to determine if it holds true.\(2(-1) - 3(2) = -2 - 6 = -8\).Since the left-hand side equals the right-hand side, the third equation holds true.

Conclusion

Since all provided equations hold true after substituting \(x = 1\), \(y = -1\), and \(z = 2\), the values \((1, -1, 2)\) are indeed solutions to the system of equations.

Key concepts

Substitution Method

The substitution method is a widely-used technique for solving systems of equations.
It involves replacing one variable in an equation with its equivalent from another equation.
This helps in simplifying the system to make solving easier. Let's break it down:

First, you solve one of the equations for one of the variables. For instance, if the system includes \( x + y = 10 \) and \( x - y = 2 \), you can solve the second equation for \( x \):
\[x = y + 2\]

Next, substitute this expression in place of \(x \) in the first equation:
\[ y + 2 + y = 10 \]

Then, solve for \( y \):
\[ 2y + 2 = 10 \implies 2y = 8 \implies y = 4 \]

Finally, substitute \( y \) back into the expression for \( x \):
\[ x = 4 + 2 = 6 \]

This technique simplifies complex systems and helps in finding variables' values systematically. You substitute, simplify, and solve until you've worked through all equations.

Solution Verification

Verification of solutions is essential to confirm that the values satisfy all given equations.
It's a process of plugging the values back into each original equation and checking the results.
Here's how it's done:

Given our system of equations:
\[ \begin{array}{l} 3x + 3y + 2z = 4,\ x - y - z = 0,\ 2y - 3z = -8 \end{array} \]
And values: (\(x = 1, y = -1, z = 2\)):

First, substitute \( x, y, \) and \( z \) into the first equation:
\[ 3(1) + 3(-1) + 2(2) = 3 - 3 + 4 = 4 \]
It holds true (4 = 4).

Then, substitute into the second equation:
\[ 1 - (-1) - 2 = 1 + 1 - 2 = 0 \]
It holds true too (0 = 0).

Finally, substitute into the third equation:
\[ 2(-1) - 3(2) = -2 - 6 = -8 \]
The equation holds true (-8 = -8).

This repetitive pattern of substitution and simplification ensures all values meet the given equations. Always verify to avoid errors.

Linear Equations

A linear equation is an equation that forms a straight line when graphed.
Each term is either a constant or a product of a constant and a single variable.
Linear equations come in different forms, such as standard form \(Ax + By = C\) and slope-intercept form \(y = mx + b\).

For example, in our system of equations:
\[ 3x + 3y + 2z = 4 \]
\[ x - y - z = 0 \]
\[ 2y - 3z = -8 \]
Each equation is linear because their variables (\(x\), \(y\), and \(z\)) are to the first power.

Key properties of linear equations include:

• No variables are multiplied together.
• No variables are squared or have higher exponents.
• The graph forms a straight line.
• They can be written in various forms but ultimately represent straight lines in multi-dimensional space.

Linear equations are foundational in algebra and crucial for understanding more advanced mathematics. They model relationships assuming constant rates and provide clear, interpretable results when graphed or solved algebraically.