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Chapter 13: Problem 15

Solve by the addition method. $$\begin{aligned} &2 x-3 y=1\\\ &4 x-6 y=2 \end{aligned}$$

Short Answer

Expert verified
The system of equations has infinitely many solutions since these two equations are dependent.

Step by step solution

01

Observe the equations

Observe that \(2x-3y=1\) and \(4x-6y=2\) are multiples of each other. Equation 2 is just 2 times the equation 1.
02

Apply Addition Method

Usually, for addition method, we add the equations together, but in this case adding them would not give a new non-trivial equation as equation 2 is just 2 times equation 1. Therefore, this system of equations is dependent.
03

Provide conclusion

Given the dependency of the equations, we therefore have infinitely many solutions to these equations. They are essentially the same line.

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Key Concepts

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

Dependent Equations
Dependent equations are systems of equations where one equation is a multiple of another. In simpler terms, dependent equations are like different expressions of the same equation.
This means that instead of having two independent paths or lines, these equations overlap completely. They tell us the same thing in different forms.
So, in our exercise, since the second equation is exactly twice the first, they don't provide additional solving information. When you encounter dependent equations:
  • Recognize them by seeing if one is a multiple of another.
  • This dependency implies both equations describe the same line, so solving them will yield relationships that apply to both.
Understanding dependency helps in predicting the type and number of solutions a system may have.
Infinite Solutions
When systems of equations are dependent, it often leads to infinite solutions. But what does that mean?
In the context of linear equations, infinite solutions mean there are countless values for the variables that satisfy both equations at the same time. This happens when the equations represent the same line on a graph.
Thus, every point on this line is a valid solution. To identify infinite solutions in a system:
  • Look for dependent equations, where one is a multiple of the other.
  • Pay attention to identical simplified forms or overlapping graphs.
Knowing when a system has infinite solutions is crucial to solving it accurately.
Multiples of Equations
Identifying multiples of equations is essential when working with linear systems. Being multiples means one equation can be derived from another by multiplying every term by the same constant number.
This can happen in many scenarios, not just limited to linear equations.
For example, in our exercise:
  • Equation 2: \(4x-6y=2\) is a simple multiple of Equation 1: \(2x-3y=1\).
  • They represent the same line due to this multiplicative relationship.
Understanding this helps us see why such systems don鈥檛 provide unique solutions, as they lack the second independent piece of information needed to pinpoint a single solution.When analyzing systems of equations, check for multiples to better understand the nature of the solutions.

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Most popular questions from this chapter

Solve by substitution. \(\begin{aligned} y &=3 \\ 3 x-2 y &=6 \end{aligned}\)

Rewrite equation so that the coefficients are integers. Then solve the system of equations by the substitution method. \(1.2 x+0.1 y=1.9\) \(0.1 x+0.3 y=2.2\)

A police officer has chosen a high-yield stock fund that earns \(8 \%\) annual simple interest for part of a \(\$ 6000\) investment. The remaining portion is used to purchase a preferred stock that earns \(11 \%\) annual simple interest. How much should be invested in each account so that the amount earned on the \(8 \%\) account is twice the amount earned on the \(11 \%\) account?

A, B, C,\( and \)D$ are nonzero real numbers. State whether the system of equations is independent, inconsistent, or dependent. $$\begin{aligned} &x=C\\\ &y=D \end{aligned}$$

Solve by substitution. \(\begin{aligned} x-4 y &=9 \\ 2 x-3 y &=11 \end{aligned}\)
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