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Chapter 4: Problem 38

Solve each system. $$\begin{aligned}2 x-y &=6 \\\2 y+z &=-4 \\\8 x &=3\end{aligned}$$

Short Answer

Expert verified
x = \frac{3}{8}, y = -\frac{21}{4}, z = \frac{13}{2}

Step by step solution

01

- Solve for x from the third equation

The third equation is: \[ 8x = 3 \]Divide both sides by 8 to isolate x: \[ x = \frac{3}{8} \]
02

- Solve for y using the first equation

Substitute \( x = \frac{3}{8} \) into the first equation: \[2 \left( \frac{3}{8} \right) - y = 6 \]Simplify and solve for y: \[ \frac{3}{4} - y = 6 \]Subtract \( \frac{3}{4} \) from both sides: \[ -y = 6 - \frac{3}{4} \]Combine the fractions on the right: \[ -y = \frac{24}{4} - \frac{3}{4} = \frac{21}{4} \]Multiply both sides by -1 to solve for y: \[ y = -\frac{21}{4} \]
03

- Solve for z using the second equation

Substitute \( y = - \frac{21}{4} \) into the second equation: \[ 2y + z = -4 \]\[ 2 \left( - \frac{21}{4} \right) + z = -4 \]Simplify and solve for z: \[ - \frac{42}{4} + z = -4 \]\[ - \frac{21}{2} + z = -4 \]Add \( \frac{21}{2} \) to both sides: \[ z = -4 + \frac{21}{2} \]Convert \(-4\) into a fraction: \[ z = - \frac{8}{2} + \frac{21}{2} \]Combine the fractions: \[ z = \frac{13}{2} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

linear equations
A linear equation is an equation that makes a straight line when plotted on a graph. It usually takes the form:
  • \(ax + by = c\) which means you only have variables raised to the power of 1;
  • no variables multiplied together;
  • and no variables in denominators.
In our given problem, we are dealing with a system of linear equations:
  • \(2x - y = 6\);
  • \(2y + z = -4\);
  • \(8x = 3\).
Each of these equations represents a plane in 3D space. Solving the system means finding the point where these planes intersect.
substitution method
The substitution method is a way to solve a system of equations by solving one of the equations for one variable, and then substituting that expression into the other equations. Here鈥檚 how it works step-by-step with our example:
First, solve the third equation for \(x\):
  • \(8x = 3\)
  • \(x = \frac{3}{8}\).
Next, substitute \(\frac{3}{8}\) for \(x\) in the first equation:
  • \(2\frac{3}{8} - y = 6\);
  • \(\frac{3}{4} - y = 6\)
Solve for \(y\), then use this result in the second equation to find \(z\). This systematic approach leverages simpler substitution and successive solving to crack the problem.
fraction arithmetic
Understanding fractions is crucial when dealing with algebraic equations, especially in systems like ours where solutions include fractions. Here are some tips:
To simplify:
  • \(\frac{3}{8}\) stays as is since it's already simplified.
To handle addition and subtraction, find a common denominator:
  • For \(\frac{24}{4} - \frac{3}{4}\), the common denominator is 4, so it becomes \(\frac{24-3}{4} = \frac{21}{4}\).
When moving terms in an equation like \(y = -\frac{21}{4}\), multiply by -1 to switch signs. Finally, adding fractions involves the same principle of a common denominator, seen in combining \(\frac{21}{2}\) and \(-\frac{8}{2}\) to form \(-4 + \frac{21}{2} = \frac{13}{2}\).
solving for variables
Solving for variables means isolating a variable on one side of the equation to find its value. Here鈥檚 how to do it in our problem:
  • From \(8x = 3\), dividing by 8 isolates x: \(x = \frac{3}{8}\).
  • For y, substitute x in the first equation, simplify and isolate: \(-\frac{21}{4}\) gives \(y = -\frac{21}{4}\).
  • For z, substitute y into the second equation, simplify and isolate: \(\frac{13}{2}\) gives \(z = \frac{13}{2}\).
The key is logical steps: simplify, substitute, and solve sequentially to peel away layers of the problem.

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Most popular questions from this chapter

Solve each system using the Gauss-Jordan elimination method. $$ \begin{array}{l} 3 x-4 y=-1 \\ x-y=0 \end{array} $$

Use Cramer's rule to solve each system. \(\begin{aligned} x+y+z &=0 \\ 2 y+2 z &=0 \\ 3 x-y &=-1 \end{aligned}\)

Solve each problem by using a system of three equations in three unknowns. Working overtime. To make ends meet, Ms. Farns by works three jobs. Her total income last year was 48,000 dollars. Her income from teaching was just 6000 dollars more than her income from house painting. Royalties from her textbook sales were one-seventh of the total money she received from teaching and house painting. How much did she make from each source last year?

Solve each system using the Gauss-Jordan elimination method. $$ \begin{array}{r} x-3 y=8 \\ 2 x-6 y=1 \end{array} $$

Determine the size of each matrix. $$ \left[\begin{array}{llll} 0 & 0 & 0 & 6 \end{array}\right] $$
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