Question

Solve the system of equations. $$\left\\{\begin{array}{rr}x+4 z & =13 \\ 4 x-2 y+z= & 7 \\ 2 x-2 y-7 z & =-19\end{array}\right.$$

Short answer

The solution of the system of equations is x = 13, y = 22.5, and z = 0.

Step-by-step solution

Simplify the equations

The first step is to make the equations easier to manage. Let's isolate 'x' in the first equation.\[x = 13 - 4z\]

Substitute x in the other equations

Substitute the obtained value of 'x' from step 1 into the remaining two equations:\[4(13 - 4z) - 2y + z = 7\]\[2(13 - 4z) - 2y - 7z = -19\]Now simplify these equations and isolate y. After simplifying, we get:\[-6z - 2y = -45\]\[-15z - 2y = -45\]This gives us a system of two new equations with two variables (y and z).

Solve the system of two equations

To find the values of 'y' and 'z', we can subtract the second equation from the first:\[9z = 0\]Solving this, we find z = 0. Substituting 'z' back into one of the two-variable equations, we can find 'y'. Let's use the first equation:\[-6(0) - 2y = -45\]Solving this equation, we find y = 22.5.

Find the value of 'x'

Substitute the values of 'y' and 'z' into the original first equation to find 'x':\[x = 13 - 4(0)\]So, x = 13.

Key concepts

Understanding Algebraic Solutions

An algebraic solution involves finding the values of variables that satisfy given equations. In this context, we are solving a system of linear equations, which consists of multiple equations that are related through common variables. The goal is to find a set of values for the variables that make all the equations true at the same time.

To achieve this, we manipulate the equations using policies such as substitution or elimination, eventually finding a single solution or determining that no solution exists. Each equation in the system forms a straight line when graphed on a coordinate plane, and the solution is the point where these lines intersect, representing the common values of the variables.

• The solution involves simplifying equations by eliminating variables systematically.
• We strive to get each equation in terms of a single variable to solve them one by one, which facilitates tracking and solving simultaneously multiple variables.

Exploring the Substitution Method

The substitution method is one of the core approaches to solving systems of equations. It involves expressing one variable in terms of the others and substituting this expression into the other equations in the system.

In our specific example, we started by solving for 'x' in the first equation. By isolating 'x', we get: \[ x = 13 - 4z \]. This expression for 'x' is then placed into the other equations, simplifying the system to focus on the remaining variables, in this case, 'y' and 'z'. This method is particularly useful when one of the equations is easily solvable for one of the variables, as it simplifies the process of dealing with more complex systems.

• Prioritize isolating a variable that simplifies the procedure.
• Efficiency increases with simpler substitution equations.

The Basics of Linear Equations

Linear equations form the basis of the system we are solving. A linear equation is any equation that graphs to a straight line and is formed by constants and variables raised to the first power.

In a system of linear equations, we are looking for a point that simultaneously lies on all the lines represented by these equations. This system might have one solution, no solution, or infinitely many solutions. In our exercise, we found only one solution, indicating the lines intersect precisely at one point in space.

• Each linear equation expresses a relationship between variables.
• The simplicity of linear equations allows straightforward graphical interpretation.