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Chapter 8: Problem 4

In Exercises \(1-8\), take a trip down memory lane and solve the given system using substitution and/or elimination. Classify each system as consistent independent, consistent dependent, or inconsistent. Check your answers both algebraically and graphically. $$ \left\\{\begin{array}{l} \frac{2}{3} x-\frac{1}{5} y=3 \\ \frac{1}{2} x+\frac{3}{4} y=1 \end{array}\right. $$

Short Answer

Expert verified
The system is consistent and independent with the solution \( (x, y) = \left( \frac{49}{12}, -\frac{25}{18} \right) \).

Step by step solution

01

Elimination Preparation

Begin by ensuring that the coefficients of one of the variables are the same in both equations. Consider multiplying the first equation by 5 and the second equation by 2 to eliminate the fractions and facilitate the elimination process.
02

Simplify Equations

After multiplying, simplify each equation: \( 5 \cdot \frac{2}{3}x - 5 \cdot \frac{1}{5}y = 5 \cdot 3 \) becomes \( \frac{10}{3}x - y = 15 \) and \( 2 \cdot \frac{1}{2}x + 2 \cdot \frac{3}{4}y = 2 \cdot 1 \) becomes \( x + \frac{3}{2}y = 2 \).
03

Fraction Elimination

To eliminate the fractions, multiply the first equation by 3 and the second by 2: \( 10x - 3y = 45 \) and \( 2x + 3y = 4 \).
04

Elimination Process

Add the two equations to eliminate \( y \): \( 10x - 3y + 2x + 3y = 45 + 4 \), which simplifies to \( 12x = 49 \).
05

Solve for \( x \)

Solve the equation \( 12x = 49 \) for \( x \) by dividing both sides by 12: \( x = \frac{49}{12} \).
06

Substitute to Find \( y \)

Use the expression for \( x \) in either of the simplified equations. Substituting \( x = \frac{49}{12} \) into \( x + \frac{3}{2}y = 2 \): \( \frac{49}{12} + \frac{3}{2}y = 2 \).
07

Solve for \( y \)

Re-arrange the equation to find \( y \): \( \frac{3}{2}y = 2 - \frac{49}{12} \), calculating \( 2 = \frac{24}{12} \) gives \( \frac{3}{2}y = \frac{24}{12} - \frac{49}{12} = -\frac{25}{12} \). Therefore, \( y = \frac{-25}{12} \times \frac{2}{3} = -\frac{25}{18} \).
08

Classification and Verification

The solution \( (x, y) = \left( \frac{49}{12}, -\frac{25}{18} \right) \) makes the system consistent and independent, as we have a unique solution. Verify by substituting \( x \) and \( y \) back into the original equations and checking for equality. Graphically, check that the two lines intersect at \( \left( \frac{49}{12}, -\frac{25}{18} \right) \).

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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 handy technique used to solve systems of equations. It works by expressing one variable in terms of the other and substituting this relation into the other equation. This approach is particularly useful when one of the equations is already solved for a variable.

### Steps to Apply Substitution Method
  • Solve one of the equations for one of the variables. For example, if you have two equations, try to express one variable, like \( y \), in terms of \( x \).
  • Substitute this expression into the other equation. This substitution will replace the variable and give you an equation with only one variable, which you can solve.
  • Find the value of the other variable by substituting the value obtained back into the equation solved initially.
This method is especially beneficial when dealing with simple linear equations without complex coefficients.
Elimination Method
The elimination method, also known as the addition method, is another fascinating way to tackle systems of equations. This process involves adding or subtracting equations to eliminate one of the variables.

### Steps for Using Elimination Method
  • Multiply one or both of the equations by a constant so that when you add or subtract them, one of the variables cancels out. This often involves creating equal coefficients for one variable in both equations.
  • Once a variable is eliminated, solve for the remaining variable. This leaves you with a single equation, which is more straightforward to solve.
  • Substitute the value of the solved variable back into one of the original equations to find the value of the other variable.
In our original exercise, we used elimination by aligning the coefficients and then adding the two equations to remove the \( y \) term, eventually solving for \( x \).
Consistent and Independent Systems
In the context of systems of equations, it is crucial to classify the system to understand its solutions better. A system is described as consistent and independent if it has exactly one solution. This generally means the lines represented by the equations intersect at a single point on the graph.

### Characteristics of Consistent and Independent Systems
  • Unique Solution: Only one pair of \( x \) and \( y \) satisfies both equations.
  • Intersection Point: Graphically, the lines cross exactly once.
  • Non-parallel: The lines are not parallel, ensuring they meet at some point.
In our exercise, the system given is consistent and independent as evidenced by the calculation yielding a unique solution \( (x, y) = \left( \frac{49}{12}, -\frac{25}{18} \right) \).
Algebraic Verification
Algebraic verification is a method to confirm the solution obtained from solving a system of equations. Once you have a potential solution, you substitute these values back into the original equations to verify their correctness.

### Steps for Algebraic Verification
  • Substitute the solution for \( x \) and \( y \) into both original equations.
  • Check if both sides of the equations are equal with the substituted values. If both equations are satisfied, the solution is verified.
This step is crucial for confirming the accuracy of the solution and avoiding errors. In our exercise, substituting \( (x, y) = \left( \frac{49}{12}, -\frac{25}{18} \right) \) confirms both original equations, ensuring the solution is correct.
Graphical Verification
Graphical verification provides a visual confirmation of the solution by plotting the equations on a graph. This technique allows us to see the solutions as intersection points of the lines.

### Steps to Perform Graphical Verification
  • Convert each equation into a form that can be easily graphed, like the slope-intercept form \( y = mx + b \).
  • Plot both equations on the same graph to identify the point of intersection.
  • Check that the intersection point matches the algebraically obtained solution.
By checking where the lines intersect, graphical verification visually confirms the system's solution. In the exercise, the lines meet at \( \left( \frac{49}{12}, -\frac{25}{18} \right) \), proving the solution's validity.

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Most popular questions from this chapter

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