Question

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{aligned} x+y &=9 \\ 2 x &-z=13 \\ 3 y+2 z &=7 \end{aligned}\right. $$

Short answer

Solution: \( x = \frac{46}{7}, y = \frac{17}{7}, z = -\frac{1}{7} \). Verified.

Step-by-step solution

Express Variables in Terms of One Another

First, take the first equation, which is \( x + y = 9 \), and solve for one of the variables. Let's solve for \( x \): \[ x = 9 - y \]

Substitute into the Second Equation

Next, substitute \( x = 9 - y \) into the second equation \( 2x - z = 13 \): \[ 2(9 - y) - z = 13 \] Simplify this equation: \[ 18 - 2y - z = 13 \] \[ -2y - z = -5 \] \[ z = -5 + 2y \]

Substitute into the Third Equation

Take \( z = -5 + 2y \) and substitute into the third equation \( 3y + 2z = 7 \): \[ 3y + 2(-5 + 2y) = 7 \] Simplify this equation: \[ 3y - 10 + 4y = 7 \] \[ 7y - 10 = 7 \] \[ 7y = 17 \] \[ y = \frac{17}{7} \]

Find \( x \)

Now that \( y = \frac{17}{7} \), substitute back into \( x = 9 - y \): \[ x = 9 - \frac{17}{7} \] Convert 9 to a fraction: \[ x = \frac{63}{7} - \frac{17}{7} \] \[ x = \frac{46}{7} \]

Find \( z \)

Finally, substitute \( y = \frac{17}{7} \) into \( z = -5 + 2y \): \[ z = -5 + 2 \times \frac{17}{7} \] Convert -5 to a fraction: \[ z = -\frac{35}{7} + \frac{34}{7} \] \[ z = -\frac{1}{7} \]

Verify the Solution

Verify the solution by substituting \( x, y, \) and \( z \) back into the original equations to ensure they hold true. For \( x + y = 9 \): \[ \frac{46}{7} + \frac{17}{7} = 9 \] True. For \( 2x - z = 13 \): \[ 2 \times \frac{46}{7} - (-\frac{1}{7}) = 13 \] True. For \( 3y + 2z = 7 \): \[ 3 \times \frac{17}{7} + 2 \times (-\frac{1}{7}) = 7 \] True.

Key concepts

Linear Equations

Linear equations are mathematical expressions that represent straight lines on a graph. In a system of linear equations, multiple lines interact with each other. Each equation represents a line with a specific slope and y-intercept.

For example, the first equation in the given problem is: \( x + y = 9 \). This means every point on this line adds up to 9 when the values of \( x \) and \( y \) are combined.

Understanding these linear equations is crucial because they are the building blocks of the entire system. By graphing these lines, you can visually interpret their intersection points, which represent the solutions to the system.

Key points to remember:

• A linear equation can be written in the form \( ax + by = c \), where \( a, b, \) and \( c \) are constants.
• The solution to a system of linear equations is the point where all lines intersect.
• It's possible for a system to have one solution, no solution, or infinitely many solutions.

Substitution Method

The substitution method is a common technique for solving systems of equations. It involves solving one of the equations for one variable and then substituting that expression into the other equations.

In the given problem, we start by solving the first equation \( x + y = 9 \) for \( x \):

\( x = 9 - y \).

Then, we substitute this into the second equation to find an expression for \( z \):

\( 2(9 - y) - z = 13 \)

Simplifying this gives us \( z = -5 + 2y \).

Finally, we substitute \( z \) into the third equation to solve for \( y \):

\( 3y + 2(-5 + 2y) = 7 \).

The substitution method can require a lot of algebraic manipulation, but it is powerful because it systematically reduces the number of variables, simplifying the problem step by step.

Important points:

• Always start by isolating one variable in one of the equations.
• Substitute the expression into the other equations to reduce the number of variables.
• Solve the resulting equations step by step until you find the values of all variables.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to isolate variables and solve for their values. It is an essential skill when solving systems of equations.

In the given problem, we use algebraic manipulation multiple times:

1. Simplifying \( 2(9 - y) - z = 13 \) to find \( z \):

\( 18 - 2y - z = 13 \)

\( -2y - z = -5 \)

\( z = -5 + 2y \).

2. Solving for \( y \) in the modified third equation:

\( 3y + 2z = 7 \)

Substituting \( z = -5 + 2y \)

\( 3y + 2(-5 + 2y) = 7 \)

\( 3y - 10 + 4y = 7 \)

\( 7y - 10 = 7 \)

\( y = \frac{17}{7} \).

3. Finding \( x \) and \( z \) once \( y \) is determined:

\( x = 9 - \frac{17}{7} \)

\( x = \frac{46}{7} \).

\( z = -5 + 2 \times \frac{17}{7} \)

\( z = -\frac{1}{7} \).

Critical aspects:

• Perform operations equally on both sides of the equation to maintain balance.
• Always simplify equations as much as possible before substituting values.
• Double-check your work by substituting the found values back into the original equations to verify their correctness.