Question

Solve the simultaneous equations $$ \begin{aligned} 2 x+3 y+z &=11 \\ x+y+z &=6 \\ 5 x-y+10 z &=34 \end{aligned} $$

Short answer

x = 3, y = 1, z = 2

Step-by-step solution

Write the system of equations

The system of equations is:\[ \begin{aligned} 2x + 3y + z &= 11 \ x + y + z &= 6 \ 5x - y + 10z &= 34 \end{aligned} \]

Eliminate one variable (choose z)

Subtract the second equation from the first equation to eliminate \(z\):\[ (2x + 3y + z) - (x + y + z) = 11 - 6 \ x + 2y = 5 \]

Eliminate the same variable (z) from another pair

Multiply the second equation by 10 and subtract it from the third equation:\[ (5x - y + 10z) - 10(x + y + z) = 34 - 60 \ 5x - y + 10z - 10x - 10y - 10z = -26 \ -5x - 11y = -26 \ -5x - 11y = -26 \ 5x + 11y = 26 \]

Solve the two-variable system

We have:\[ x + 2y = 5 \ 5x + 11y = 26 \]Solve for \(x\) and \(y\) using substitution or elimination. Multiply the first equation by 5 and subtract from the second:\[ 5(x + 2y) = 5(5) \ 5x + 10y = 25 \ 5x + 11y = 26 - (5x + 10y = 25) \ 5x + 11y - 5x - 10y = 26 - 25 \ y = 1 \]

Substitute back to find x

Substitute \(y = 1\) into \(x + 2y = 5\):\[ x + 2(1) = 5 \ x + 2 = 5 \ x = 3 \]

Substitute x and y to find z

Substitute \(x = 3\) and \(y = 1\) into \(x + y + z = 6\):\[ 3 + 1 + z = 6 \ z = 6 - 4 \ z = 2 \]

Verify the solution

Substitute \(x = 3\), \(y = 1\), and \(z = 2\) back into the original equations to verify:\[ 2(3) + 3(1) + 2 = 6 + 3 + 2 = 11 \] which is correct.\[ 3 + 1 + 2 = 6 \] which is correct.\[ 5(3) - 1 + 10(2) = 15 - 1 + 20 = 34 \] which is correct.

Key concepts

System of Linear Equations

A system of linear equations consists of two or more equations that share the same set of variables. These equations represent lines in a multi-dimensional space.

In our given problem, we have three linear equations with three variables:

• \[ 2x + 3y + z = 11 \]
• \[ x + y + z = 6 \]
• \[ 5x - y + 10z = 34 \]

Our task is to find the values of x, y, and z that satisfy all three equations simultaneously.

To solve these, we use methods like variable elimination and substitution, which simplify the system step-by-step. In this way, we convert a complex system into simpler terms to find the solution for each variable.

Variable Elimination

Variable elimination aims to reduce the number of variables in the system step-by-step. This process makes the equations easier to solve.

Let's start by eliminating the variable z from our equations. First, we subtract the second equation from the first:

• \[ (2x + 3y + z) - (x + y + z) = 11 - 6 \]
• \[ x + 2y = 5 \]

This gives us a new equation with only x and y. Next, we eliminate z again but from a different pair of equations. We multiply the second equation by 10 and then subtract it from the third equation:

• \[ (5x - y + 10z) - 10(x + y + z) = 34 - 60 \]
• \[ 5x - y + 10z - 10x - 10y - 10z = -26 \]
• \[ -5x - 11y = -26 \]
• \[ 5x + 11y = 26 \]

Now, we have two new simplified equations, x + 2y = 5 and 5x + 11y = 26, which only involve x and y.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting this value into another equation. This helps in finding the value of other variables.

From our simplified system, we solve one equation for one of the variables. Let's use the first equation, x + 2y = 5. We isolate x:

• \[ x = 5 - 2y \]

We then substitute this into the second simplified equation, 5x + 11y = 26:

• \[ 5(5 - 2y) + 11y = 26 \]
• \[ 25 - 10y + 11y = 26 \]
• \[ 25 + y = 26 \]
• \[ y = 1 \]

Once y is known, substitute y = 1 back into the first simplified equation to find x:

• \[ x + 2(1) = 5 \]
• \[ x + 2 = 5 \]
• \[ x = 3 \]

Now we have x = 3 and y = 1.

Verification of Solutions

After finding the values of the variables, it's crucial to verify that these values satisfy all the original equations. Substitute x = 3, y = 1, and z = 2 into the original system:

• \[ 2(3) + 3(1) + 2 = 6 + 3 + 2 = 11 \] - This is correct.
• \[ 3 + 1 + 2 = 6 \] - This is correct.
• \[ 5(3) - 1 + 10(2) = 15 - 1 + 20 = 34 \] - This is also correct.

All three equations are satisfied, which confirms that our solutions x = 3, y = 1, and z = 2 are correct. Verification is an essential step in solving any system of equations, ensuring that our solutions are accurate.