Problem 79 Solve. If the equation has no so... [FREE SOLUTION]

Chapter 5: Problem 79

Solve. If the equation has no solution, write "No solution." $$\frac{2(5 x-6)-3(x-4)}{7}=x+2$$

Short Answer

Expert verified

The resulting equation $0 = 2$ is always false, indicating there's no $x$ that would satisfy the original equation. Therefore the answer is 'No solution'.

Step by step solution

Apply Distributive Property

Let's start with the left side of the equation, apply the distributive property to the numerator to get rid of the brackets. This gives $ \frac{2(5x-6)-3(x-4)}{7} = \frac{10x-12-3x+12}{7} $

Simplify the Numerator and denominator

The numerator simplifies to $10x - 3x = 7x$, and the denominator remains as 7. This gives us a new equation: $\frac{7x}{7} = x + 2 $

Solve for $x$

Now, cancel out the 7 from the numerator and denominator on the left side, which simplifies the equation to: $x = x + 2$

Check for valid $x$

Subtract $x$ from both sides to see if there's a valid $x$ that would satisfy the equation: $0 = 2$

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Key Concepts

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

Distributive Property

The distributive property is a fundamental algebraic principle used to simplify expressions. It allows you to multiply a single term by terms inside a parenthesis. Here's how it works: you multiply the term outside by each term inside the parenthesis.
For example, in the problem from our exercise: $2(5x - 6)$, you'll distribute the 2 across both $5x$ and $-6$. This results in $10x - 12$.
Similarly, you'll do the same with $-3(x - 4)$, which will become $-3x + 12$.

The goal of using the distributive property here is to remove the parentheses by spreading out the multiplication over addition or subtraction inside the parentheses.
Be careful with signs; distributing a negative sign will change the signs of terms inside the parentheses.

Understanding and applied correctly, the distributive property helps set up equations for further simplification.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms and ensuring the equation is easier to work with. After distributing in our problem, you have terms like $10x$ and $-3x$ in the expression $10x - 3x + (-12 + 12)$.
Combine these terms:

Like terms are terms that contain the same variable. In this case, you can combine $10x$ and $-3x$ into $7x$.
Notice how the other terms $-12$ and $+12$ cancel each other out, simplifying the equation further.

This process helps dismantle complex-looking algebra into simpler, more manageable expressions. Once simplified, it's much easier to spot the nature of the equation, such as whether it has a solution or not.

Equations with No Solution

In algebra, an equation with no solution means that no value for the variable will satisfy the equation. In our exercise, after simplifying the expression to $x = x + 2$, an inspection shows a contradiction.

We try isolating terms by subtracting $x$ from both sides, resulting in the absurd statement $0 = 2$.
This indicates there's no possible value of $x$ that makes this equation true since zero can never equal two.

Such instances arise in particular types of equations and indicate inconsistency where no real solution exists. Recognizing this outcome is critical when solving, as not all equations will yield variable values to balance both sides.

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91影视

Solve. If the equation has no solution, write "No solution." $$\frac{2(5 x-6)-3(x-4)}{7}=x+2$$

Short Answer

Step by step solution

Apply Distributive Property

Simplify the Numerator and denominator

Solve for \(x\)

Check for valid \(x\)

Key Concepts

Distributive Property

Simplifying Algebraic Expressions

Equations with No Solution

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