Question

Verify that the values of the variables listed are solutions of the system of equations. $$ \begin{array}{l} \left\\{\begin{aligned} 3 x+3 y+2 z &=4 \\ x-3 y+z &=10 \\ 5 x-2 y-3 z &=8 \end{aligned}\right. \\ x=2, y=-2, z=2 ;(2,-2,2) \end{array} $$

Short answer

Yes, (2, -2, 2) is a solution.

Step-by-step solution

Substitute the values into the first equation

Substitute the given values of the variables, \( x = 2 \), \( y = -2 \), and \( z = 2 \), into the first equation: 3x + 3y + 2z = 4 Calculating the left side: \( 3(2) + 3(-2) + 2(2) = 6 - 6 + 4 = 4 \) The left side equals the right side, 4, so the value satisfies the first equation.

Substitute the values into the second equation

Substitute \( x = 2 \), \( y = -2 \), and \( z = 2 \) into the second equation: x - 3y + z = 10 Calculating the left side: \( 2 - 3(-2) + 2 = 2 + 6 + 2 = 10 \) Again, the left side equals the right side, 10, so the value satisfies the second equation.

Substitute the values into the third equation

Substitute \( x = 2 \), \( y = -2 \), and \( z = 2 \) into the third equation: 5x - 2y - 3z = 8 Calculating the left side: \( 5(2) - 2(-2) - 3(2) = 10 + 4 - 6 = 8 \) The left side equals the right side, 8, so the value satisfies the third equation.

Key concepts

Substitution Method

The substitution method is a way to solve systems of equations by solving one of the equations for one variable and substituting this expression into the other equations. This helps to reduce the number of variables and equations step-by-step.

To use the substitution method, follow these steps:

• Choose one of the equations and solve it for one of the variables.
• Substitute this expression into the other equations.
• Solve the new equations for the remaining variables.
• Use the solved variable to find the value of the first variable if needed.

In our example, instead of solving for individual variables first, we are directly given values to verify. This also indirectly employs substitution to check if the values keep all the equations balanced on both sides.

Verifying Solutions

Verifying solutions ensures that the given values indeed satisfy each equation in the system. In simple terms, we substitute the values back into the equations and check if the left-hand side equals the right-hand side.

• Substitute the given values into the first equation.
• Calculate to see if both sides of the equation match.
• Repeat the process for the second and third equations.

Using our exercise as an example:

First equation: 3x + 3y + 2z = 4
Substituting x=2, y=-2, z=2:
3(2) + 3(-2) + 2(2) = 6 - 6 + 4 = 4
Since the left side equals the right side, the first equation is satisfied.

Similar steps follow for the second and third equations. Ultimately, all given values should balance each equation, confirming they are correct solutions.

Linear Equations

Linear equations are equations of the first degree, meaning they involve only the first power of the variables with no exponents. They represent straight lines when graphed on a coordinate plane.

Basic properties of linear equations include:

• They have one or more variables.
• The highest power of the variable is 1.
• They produce a straight line graph.

Linear equations in a system can be solved using methods like substitution or elimination to find the point(s) of intersection. In our exercise, we have a system of three linear equations. By substituting given values and verifying them, we check if these values pinpoint a common intersection point across all three linear relationships.

For equations such as:
3x + 3y + 2z = 4
x - 3y + z = 10
5x - 2y - 3z = 8
We see these as planes in a three-dimensional space. Solving the system means locating the exact (x, y, z) coordinates where all three planes intersect, if they indeed do.