Question

Solve each system of equations using any method you wish. \(\left\\{\begin{aligned} x-2 y+4 z &=2 \\\\-3 x+5 y-2 z &=17 \\ 4 x-3 y &=-22 \end{aligned}\right.\)

Short answer

The solution is \[ x = -4, y = 2, z = 2.5 \] and it satisfies all original equations.

Step-by-step solution

- Identify the system of equations

The given system of equations is: 1) \[ x - 2y + 4z = 2 \] 2) \[ -3x + 5y - 2z = 17 \] 3) \[ 4x - 3y = -22 \]

- Solve for one variable

We will solve the third equation, \[ 4x - 3y = -22 \], for one of the variables. Let's solve for \[ x \]:\[ x = \frac{3y - 22}{4} \]

- Substitute into other equations

Substitute \[ x = \frac{3y - 22}{4} \] into the first and second equations.For the first equation: \[ \frac{3y - 22}{4} - 2y + 4z = 2 \]For the second equation: \[ -3(\frac{3y - 22}{4}) + 5y - 2z = 17 \]

- Simplify the substituted equations

Simplify the first substituted equation: \[ \frac{3y - 22}{4} - 2y + 4z = 2 \]\[ \frac{3y - 22 - 8y + 16z}{4} = 2 \]\[ -5y + 3 + 16z = 8 \]\[ -5y + 16z = 30 \]Simplify the second substituted equation: \[ -3(\frac{3y - 22}{4}) + 5y - 2z = 17 \]\[ \frac{-9y + 66}{4} + 5y - 2z = 17 \]\[ -9y + 66 + 20y - 8z = 68 \]\[ 11y - 8z = 2 \]

- Solve the simplified system

The new system of equations is:1) \[ -5y + 16z = 30 \]2) \[ 11y - 8z = 2 \]Multiply the first equation by 8 and the second equation by 16 to eliminate \[ z \]:\[ -40y + 128z = 240 \]\[ 176y - 128z = 32 \]Now add the two equations:\[ 136y = 272 \]\[ y = 2 \]

- Substitute \( y = 2 \) back into equations

Substitute \[ y = 2 \] back into one of the simplified equations to find \[ z \]:\[ -5(2) + 16z = 30 \]\[ -10 + 16z = 30 \]\[ 16z = 40 \]\[ z = \frac{40}{16} = \frac{5}{2} = 2.5 \]

- Find \( x \)

Now use \[ x = \frac{3y - 22}{4} \] with \[ y = 2 \] to find \[ x \]:\[ x = \frac{3(2) - 22}{4} = \frac{6 - 22}{4} = \frac{-16}{4} = -4 \]

- Solution Verification

Verify the solution by substituting \[ x = -4, y = 2, z = 2.5 \] back into the original equations:1) \[ x - 2y + 4z = -4 - 4 + 10 = 2 \] (True)2) \[ -3x + 5y - 2z = 12 + 10 - 5 = 17 \] (True)3) \[ 4x - 3y = -16 - 6 = -22 \] (True)The solution is correct.

Key concepts

substitution method

The substitution method is a popular technique for solving systems of equations. It involves solving one of the equations for a single variable and then substituting this expression into the other equations. This allows us to reduce the system to a single equation with one variable, which is usually easier to solve.

For example, in the given system of equations:
l \( x - 2y + 4z = 2 \)
l \( -3x + 5y - 2z = 17 \)
l \( 4x - 3y = -22 \),

we first isolate one variable. We choose the third equation, solve for \( x \) as \( x = \frac{3y - 22}{4} \). Next, this expression for \( x \) is substituted into the first and second equations. This creates new equations that only contain \( y \) and \( z \). After simplifying, we end up with a new system that's easier to solve:
l \( -5y + 16z = 30 \)
l \( 11y - 8z = 2 \).

The purpose of using substitution is to reduce complexity. By transforming a more complicated system into simpler equations, we can use basic algebra to find the solution.

linear equations

Linear equations are equations of the first degree, meaning they have variables raised only to the power of one. They can be written in the general form \( ax + by + cz = d \), where \( a \), \( b \), \( c \), and \( d \) are constants.

In the given problem, we deal with three linear equations:
l \( x - 2y + 4z = 2 \)
l \( -3x + 5y - 2z = 17 \)
l \( 4x - 3y = -22 \).

These equations represent planes in three-dimensional space. The solution to the system of linear equations is the point where these three planes intersect. To find this intersection point, we can employ methods such as substitution, elimination, or matrix operations.

Understanding linear equations is crucial because they form the basis for more advanced topics in mathematics and engineering. Their properties are helpful in solving complex problems in diverse fields such as physics, computer science, and economics.

solving systems

Solving systems of equations involves finding a set of values for the variables that satisfies all the given equations simultaneously. Systems can be solved through various methods, including substitution, elimination, and using matrices.

In our example, we used the substitution method to solve the system:
l \( x - 2y + 4z = 2 \)
l \( -3x + 5y - 2z = 17 \)
l \( 4x - 3y = -22 \).

Steps to solve using substitution are:

• Solve one equation for a single variable.
• Substitute this expression into the other equations to obtain a simpler system.
• Solve the simpler system for the remaining variables.
• Back-substitute to find the values of other variables.
• Check your solution by substituting back into the original equations.

In our case, solving \( 4x - 3y = -22 \) for \( x \) and substituting into the other equations resulted in simpler equations \( -5y + 16z = 30 \) and \( 11y - 8z = 2 \). This enabled us to solve for \( y \) and \( z \), and eventually find all three variables \( x, y, \text{ and } z \).

This systematic approach makes solving complex systems manageable and ensures a clear solution path.