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Chapter 4: Problem 36

(a) solve by elimination. (b) if there is one solution, check. $$ \begin{aligned} x-6 y &=9 \\ -2 x+12 y &=-18 \end{aligned} $$

Short Answer

Expert verified
The system has infinitely many solutions, as both equations are dependent.

Step by step solution

01

Write down the system of equations

Given equations are: 1) \( x - 6y = 9 \) 2) \( -2x + 12y = -18 \)
02

Multiply the first equation to align coefficients

Multiply the first equation by 2 to align the coefficients of \( x \):2) \[ 2(x - 6y) = 2(9) \]This gives us:\[ 2x - 12y = 18 \]
03

Add the two equations together

Add the aligned equations:\[ 2x - 12y + (-2x + 12y) = 18 + (-18) \]This simplifies to:\[ 0 = 0 \]This equation is always true and doesn't provide new information about \( x \) or \( y \).
04

Analyze the result

With the result \( 0 = 0 \), it indicates that the two original equations are dependent. Thus, the system has infinitely many solutions.
05

Check the solution

To confirm, let's rearrange the second equation:Divide the second equation by -2:\[ \frac{-2x + 12y}{-2} = \frac{-18}{-2} \]This simplifies to: \[ x - 6y = 9 \]The resulting equation is identical to the first equation, confirming that the two equations are the same.

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Key Concepts

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

Dependent Equations
In the given problem, we encountered what are known as dependent equations. Dependent equations are essentially different forms of expressing the same equation. This occurs because one equation can be derived from multiplying or rearranging the other.

For example, the system:
1) \( x - 6y = 9 \)
2) \( -2x + 12y = -18 \)
When we multiplied the first equation by 2, we got: \[ 2(x - 6y) = 2(9) \] which simplifies to: \[ 2x - 12y = 18 \]
So, both equations are actually the same when rearranged, meaning they are dependent. In such cases, they represent the same line graphically. Consequently, there's no unique solution since every point on the line is a solution.
Infinite Solutions
Since the equations are dependent, the system has infinitely many solutions. This means that any combination of \( x \) and \( y \) that satisfies one equation will satisfy the other.

Mathematically, when we simplified the system:
  • \( 2x - 12y = 18 \)
  • \( -2x + 12y = -18 \)
    • We discovered it reduces to: \( 0 = 0 \).
    This always-true equation doesn't give us any specific values for \( x \) or \( y \). It's an indication that all points (\( x, y \)) on the line \( x - 6y = 9 \) are solutions.

    So, if you have systems like this in the future, remember that dependent equations lead to infinite solutions.
Linear Equations
The original equations are linear. Linear equations are equations of the first order, which means the highest power of the variable is one. They can be written in the standard form \( ax + by = c \).

In this case, our system was:
1) \( x - 6y = 9 \)
2) \( -2x + 12y = -18 \)

These equations correspond to straight lines when graphed on a coordinate plane. For systems of linear equations, we analyze their solutions which can be:
  • One unique solution (lines intersecting at one point)
  • No solution (parallel lines)
  • Infinite solutions (coincident lines, or dependent equations)


    • In our problem, we identified they share the same line, hence infinitely many solutions.

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Most popular questions from this chapter

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