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Chapter 4: Problem 10

(a) solve by substitution. (b) if there is one solution, check. $$ \begin{aligned} &y=6 x-2 \\ &y=6 x+7 \end{aligned} $$

Short Answer

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Step by step solution

01

Set the Equations Equal

Since both equations are set equal to y, set the right-hand sides of the equations equal to each other: \[6x - 2 = 6x + 7\]
02

Solve for x

Isolate x by subtracting 6x from both sides of the equation:\[6x - 2 - 6x = 6x + 7 - 6x\] This simplifies to \[-2 = 7\]
03

Interpret the Result

The equation \(-2 = 7\) is a contradiction, meaning there is no value of x that satisfies both equations at the same time.
04

Conclusion

Since we reached a contradiction, there is no solution to this system of equations.

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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 equations. We use one equation to find the value of one variable in terms of the other variable. Then, we substitute this value into the second equation to find the final solution. For example, in the given system of equations:
$$ y = 6x - 2 $$
$$ y = 6x + 7 $$
We already have both equations equal to y. This means we can set them equal to each other:
$$ 6x - 2 = 6x + 7 $$ Using substitution, we set the right-hand side expressions equal to each other and solve for x. This simplifies the system and should ideally lead to finding the variable values that satisfy both equations.
Contradictions in Algebra
In algebra, a contradiction occurs when an equation simplifies to a false statement. This means no possible value satisfies the equation. For instance, from our substitution method, we get:
$$ 6x - 2 = 6x + 7 $$
If we subtract 6x from both sides, we end up with:
$$ -2 = 7 $$
This is clearly a false statement and a contradiction. When we encounter a contradiction while solving a system of equations, it implies something important about the system. Contradictions tell us that there鈥檚 no common solution that satisfies all given equations simultaneously.
No Solution Systems
A system of equations with no solution is called 'inconsistent.' This means the lines represented by the equations do not intersect. For example, the given system:
$$ y = 6x - 2 $$
$$ y = 6x + 7 $$
Both equations represent lines with the same slope but different y-intercepts. When extended forever, identical slopes mean the lines are parallel. Since the lines never cross, there's no point (x, y) that satisfies both equations. Thus, this system of equations has no solution.

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