Problem 73 \(-4 x+3 y=4\) \(-8 x+6 y=0\)... [FREE SOLUTION]

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Chapter 3: Problem 73

\(-4 x+3 y=4\) \(-8 x+6 y=0\)

Short Answer

Expert verified

The system of equations has no solutions.

Step by step solution

- Identify the System of Equations

The given system of equations is: 1) \(-4x + 3y = 4\)2) \(-8x + 6y = 0\)

- Observe the Coefficient Relationship

Notice that the second equation is exactly \(-2\) times the first equation. Multiply the first equation by \(-2\):1) \(-2(-4x + 3y) = -2 \times 4\)Which results in:2) \(-8x + 6y = -8\)

- Compare Equations

Now we have the transformed system:1) \(-4x + 3y = 4\)2) \(-8x + 6y = -8\)By comparing the second equation in its two forms: \(-8x + 6y = 0\) (original) \(-8x + 6y = -8\) (transformed)We observe a contradiction. These cannot both be true, so no solution exists.

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Key Concepts

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

inconsistent system

An inconsistent system of equations is a system that has no solutions. This happens when the equations represent parallel lines that never intersect. In our exercise, the system is given as follows:
-4x + 3y = 4
-8x + 6y = 0
An inconsistent system is identified by transforming the equations and noticing a contradiction. When you try to find values for x and y that satisfy both equations, you realize it's impossible.
We multiplied the first equation by -2 to get:
-8x + 6y = -8
Comparing it with the second equation:
-8x + 6y = 0
Since these two transformed equations are contradictory, we conclude that the system has no solution, rendering it inconsistent.

multiplication of equations

Multiplying equations in a system helps to detect whether they are proportional, which can simplify the comparison.
In our example, we noticed that the second equation is a multiple of the first:
-4x + 3y = 4
-8x + 6y = 0 (second equation)
To check, we multiply the first equation by -2:
-2(-4x + 3y) = -2 脳 4
This results in:
-8x + 6y = -8
This process helps to align the equations for easier comparison. It showcases whether the system is consistent, inconsistent, or dependent.

comparing equations

Comparing equations is crucial to understanding the nature of a system of linear equations. After transforming or aligning the equations using techniques like multiplication, we compare the results.
In our example:
Original system:
-4x + 3y = 4
-8x + 6y = 0
After multiplying the first equation by -2, we got:
-8x + 6y = -8
Comparison shows that the second transformed equation (-8x + 6y = -8) directly contradicts the original second equation (-8x + 6y = 0). Since both cannot be true simultaneously, the contradiction indicates that the system has no common solution.
This comparison reveals the inconsistency within the system, thereby concluding that the system is inconsistent.

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91影视

\(-4 x+3 y=4\) \(-8 x+6 y=0\)

Short Answer

Step by step solution

- Identify the System of Equations

- Observe the Coefficient Relationship

- Compare Equations

Key Concepts

inconsistent system

multiplication of equations

comparing equations

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