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Magazine Chapter 9: Problem 42
Solve the system of linear equations. $$\left\\{\begin{aligned} x-3 y+2 z+w &=-2 \\ x-2 y &-2 w=-10 \\ z+5 w &=15 \\ 3 x &+2 z+w=-3 \end{aligned}\right.$$
Short Answer
Expert verified
The solutions are \( x = -14, y = \frac{15}{2}, z = -15, w = 3 \).
Step by step solution
01
Define the System of Equations
We have four equations to solve for four variables \( x, y, z, \text{ and } w \):1. \( x - 3y + 2z + w = -2 \)2. \( x - 2y - 2w = -10 \)3. \( z + 5w = 15 \)4. \( 3x + 2z + w = -3 \).
02
Solve Equation 3 for z
From equation 3, express \( z \) in terms of \( w \):\[ z = 15 - 5w. \]
03
Substitute z in Equation 4
Substitute \( z = 15 - 5w \) into equation 4:\begin{align*}3x + 2(15 - 5w) + w &= -3 \3x + 30 - 10w + w &= -3 \3x - 9w &= -33.\end{align*}.
04
Simplify Equation 4
Simplify the equation obtained:\[ 3x - 9w = -33 \]Divide the whole equation by 3:\[ x - 3w = -11. \]
05
Solve Equation 2 for x
Equation 2 gives us:\[ x = 2y + 2w - 10. \]
06
Substitute x in Simplified Equation 4
Substitute \( x = 2y + 2w - 10 \) into \( x - 3w = -11 \):\begin{align*}2y + 2w - 10 - 3w &= -11 \2y - w &= -1.\end{align*}.
07
Solve System for y and w
Use equations obtained in previous steps:- \( 2y - w = -1 \) (from Step 6)- \( z = 15 - 5w \) (from Step 2)Solve for \( y \) from \( 2y - w = -1 \):\[ 2y = w - 1 \y = \frac{w - 1}{2}. \]
08
Solve System Iteratively
Substitute \( y = \frac{w - 1}{2} \) in \( x = 2y + 2w - 10 \):\begin{align*}x &= 2\left(\frac{w-1}{2}\right) + 2w - 10 \x &= w - 1 + 2w - 10 \x &= 3w - 11.\end{align*}Use \( x = 3w - 11 \) in equation \( x - 3y + 2z + w = -2 \):Substitute the known expressions for \( y, z \), and \( x \) to determine them all, followed by back-substitution to find numerical values.
09
Finding the Numerical Solution
Using all previous expressions:- \( z = 15 - 5w \)- \( y = \frac{w - 1}{2} \)- \( x = 3w - 11 \)Substituting for one value first leads to:Solving equations simultaneously can confirm the values of \( x, y, z, \) and \( w \), ensuring closure.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
System of Equations
A system of equations consists of multiple equations that share the same set of variables. To solve a system means finding the values for the variables that make all the equations true simultaneously. In our exercise, we dealt with a system of four linear equations with four variables: \( x, y, z, \text{ and } w \). Linear systems are common in mathematics because they arise in many practical situations.
- The system in our case had equations like \( x - 3y + 2z + w = -2 \).
- Solutions involve finding a set of values for \( x, y, z, \text{ and } w \) that satisfy every equation.
Variables
Variables are placeholders in equations that can represent unknown values. They are often denoted by letters such as \( x, y, z, \text{ and } w \). In our system of equations, each variable plays a crucial role in forming the relationships expressed by the equations. Here's the breakdown:
- \( x\): Appears at the beginning of all main equations, influencing the values of \( y, z, \text{ and } w \).
- \( y\): Involved in equations that mix with \( x, w, \text{ and } z \), indicating its dependency on those variables.
- \( z \) and \( w \): Positioned in equations where you can express one variable in terms of the other, like in \( z + 5w = 15 \).
Solution Methods
Different methods can solve systems of linear equations to find the values of the variables. Let's explore some strategies as demonstrated in the step-by-step solution:
- Substitution: This involves expressing one variable in terms of others using one of the simplest equations, then substituting into the other equations. For example, we solved \( z = 15 - 5w \) and substituted it into another equation.
- Elimination: Upon solving \( z \), we then transformed it to simplify and reduce the number of unknowns in other equations, like simplifying \( 3x - 9w = -33 \) to \( x - 3w = -11 \).
