Question

Verify that the values of the variables listed are solutions of the system of equations. $$ \begin{array}{l} \left\\{\begin{array}{l} 4 x -5 z=6 \\ 5 y-z =-17 \\ -x-6 y+5 z =24 \\ \end{array}\right.\\\ x=4, y=-3, z =2 ;(4,-3,2) \end{array} $$

Short answer

The values \((4, -3, 2)\) satisfy all three equations.

Step-by-step solution

Substitute the values into the first equation

Take the first equation \(4x - 5z = 6\) and substitute \(x = 4\) and \(z = 2\). Calculate to verify if the equation holds true.

Verify the first equation

\(4(4) - 5(2) = 16 - 10 = 6\). Since both sides of the equation are equal, the first equation is satisfied.

Substitute the values into the second equation

Take the second equation \(5y - z = -17\) and substitute \(y = -3\) and \(z = 2\). Calculate to verify if the equation holds true.

Verify the second equation

\(5(-3) - 2 = -15 - 2 = -17\). Since both sides of the equation are equal, the second equation is satisfied.

Substitute the values into the third equation

Take the third equation \(-x - 6y + 5z = 24\) and substitute \(x = 4\), \(y = -3\), and \(z = 2\). Calculate to verify if the equation holds true.

Verify the third equation

\(-4 - 6(-3) + 5(2) = -4 + 18 + 10 = 24\). Since both sides of the equation are equal, the third equation is satisfied.

Key concepts

solving systems of equations

A system of equations is a set of two or more equations with the same variables. The goal is to find values for these variables that satisfy all equations simultaneously. This concept is fundamental in algebra and is used in many fields such as engineering, physics, and economics.

To solve a system of equations, there are several methods available, such as:

• Graphing the equations
• Using the substitution method
• Using the elimination method

In our example, we verified solutions for a system of three linear equations. Solving such systems involves finding the values for the variables that make all the equations true at once.

substitution method

The substitution method is one of the most common techniques to solve a system of equations. It involves solving one of the equations for one variable in terms of others, and then substituting this expression into the other equations.

Here is a step-by-step guide using the substitution method:

• Solve one of the equations for one of the variables. This creates an expression for this variable.
• Substitute this expression into the other equation(s). This reduces the number of equations and variables.
• Solve the simplified equation for the remaining variable(s).
• Substitute the solution back into the original equation to find the value of the substituted variable.

In our example, we verified the solutions by substituting the values into each equation and checking if they hold true.

verification of solutions

Verification of solutions is a critical step in solving systems of equations. After finding potential solutions, you need to substitute these values back into the original equations to confirm that they work.

For instance, let's verify the solutions for our system:

• Substituting into the first equation: \(4x - 5z = 6\), with \(x = 4\) and \(z = 2\), we get \[4(4) - 5(2) = 16 - 10 = 6\]. This holds true.
• Substituting into the second equation: \(5y - z = -17\), with \(y = -3\) and \(z = 2\), we get \[5(-3) - 2 = -15 - 2 = -17\]. This also holds true.
• Substituting into the third equation: \(-x - 6y + 5z = 24\), with \(x = 4\), \(y = -3\), and \(z = 2\), we get \[-4 - 6(-3) + 5(2) = -4 + 18 + 10 = 24\]. This holds true as well.

All equations are satisfied, meaning the values \((x=4, y=-3, z=2)\) are indeed solutions to the system.

linear equations

Linear equations are equations of the first order. This means the variables are to the power of one. They are essential in algebra and appear in many real-world situations.

A general form of a linear equation with two variables is \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables. When plotted on a graph, these equations form straight lines.

In our example, here are the three linear equations:

• First equation: \(4x - 5z = 6\).
• Second equation: \(5y - z = -17\).
• Third equation: \(-x - 6y + 5z = 24\).

Each equation represents a plane in three-dimensional space. The solution to the system is the point where these three planes intersect. That point is \((x=4, y=-3, z=2)\) in our example.