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Chapter 4: Problem 60

Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent. $$\begin{aligned}&y=3(x-4)\\\&3 x-y=12\end{aligned}$$

Short Answer

Expert verified
The equations are dependent and have infinitely many solutions.

Step by step solution

01

- Substitute the Expression

Given the system of equations, start by substituting the expression for y from the first equation into the second equation. The first equation is \(y = 3(x-4)\). Substitute \(y\) in the second equation: \[3x - y = 12\] becomes \[3x - 3(x-4) = 12\].
02

- Simplify the Equation

Simplify the equation \[3x - 3(x-4) = 12\]. Distribute the -3: \[3x - 3x + 12 = 12\].
03

- Solve the Simplified Equation

Combine like terms: \[0x + 12 = 12\] simplifies to \[12 = 12\].
04

- Interpret the Result

Since \[12 = 12\] is a true statement and does not involve any variables, it indicates that the system has infinitely many solutions. This means the equations are dependent.

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Key Concepts

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

Substitution Method
The substitution method is a way to solve systems of linear equations. You solve one equation for one variable and then substitute that expression into the other equation. Here's how it works step by step:

Take two equations:
\(y = 3(x-4)\) and \(3x - y = 12\).

1. Solve the first equation for y.
2. Substitute this expression into the second equation.
For example, we substitute \(y = 3(x-4)\) into \(3x - y = 12\) to get \(3x - 3(x-4) = 12\).

This simplifies the system by expressing it in terms of one variable, allowing you to solve it more easily.
Dependent Systems
A dependent system of equations means that the equations are not truly independent. Instead, they describe the same line.

For example, after simplifying in our problem, we got \(3x - 3(x-4) = 12\), which simplifies to \(12 = 12\).

Since this true statement \(12 = 12\) holds for all values of x, it indicates that the equations are actually the same line in the coordinate plane. This means every solution of one equation is also a solution of the other.
Infinite Solutions
Infinite solutions occur when we have a dependent system. Since the equations are describing the same line, every point on that line is a solution to both equations.

In our example, the simplified equation turned into \(12 = 12\), which means every value of x and corresponding value of y that satisfies the first equation will automatically satisfy the second equation too.

If you ever simplify your equations and end up with a true statement without variables, you know you have infinite solutions and are dealing with a dependent system.

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