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Chapter 5: Problem 36

What is a system of linear equations in three variables?

Short Answer

Expert verified
A system of linear equations in three variables is a set of equations that use the same three variables, normally x, y, and z. They are first degree equations solved simultaneously with the solutions representing an intersection point on a three-dimensional graph.

Step by step solution

01

Definition of a 'System of Equations'

A 'system of equations' is a set or collection of equations that are solved simultaneously. The solution of a system of equations is an ordered pair (or triplet or more for more variables) that makes all equations in the system true.
02

Understanding 'Linear Equations'

Linear equations are equations of the first degree, meaning that the variables are only to the power of 1 and not subjected to functions like square roots or exponential functions. They are presented in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are real numbers.
03

The 'Three Variables'

The 'three variables' in this context are typically represented as \(x\), \(y\), and \(z\). Each variable will correspond to a dimension in a three-dimensional graph, and the system of equations will intersect at a single point in this graph.
04

Concept Integration

So, a system of linear equations in three variables is a collection of linear equations that all share the same variables \(x\), \(y\), and \(z\). The system is solved by finding the values of the variables that satisfy all the equations simultaneously.

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Most popular questions from this chapter

In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} &2 x-3 y=-13\\\ &y=2 x+7 \end{aligned} $$

Use a system of linear equations to solve Exercises \(57-67\) The graph shows the calories in some favorite fast foods. Use the information in Exercises \(57-58\) to find the exact caloric content of the specified foods. (GRAPH CAN'T COPY) A rectangular lot whose perimeter is 360 feet is fenced along three sides. An expensive fencing along the lot's length costs \(\$ 20\) per foot, and an inexpensive fencing along the two side widths costs only \(\$ 8\) per foot. The total cost of the fencing along the three sides comes to \(\$ 3280-\) What are the lot's dimensions?

In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} &y=\frac{1}{3} x+\frac{2}{3}\\\ &y=\frac{5}{7} x-2 \end{aligned} $$

In Exercises \(19-30,\) solve each system by the addition method. \(5 x=6 y+40\) \(2 y=8-3 x\)

In Exercises \(19-30,\) solve each system by the addition method. \(3 x+2 y=14\) \(3 x-2 y=10\)
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