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Chapter 1: Problem 83

Solve. $$ 7 x-2-3 x=4 x $$

Short Answer

Expert verified
There is no solution; the equation is inconsistent.

Step by step solution

01

Combine Like Terms on the Left Side

Combine the terms involving the variable x on the left-hand side of the equation. The equation is initially given as: 7x - 2 - 3x = 4x Combine the terms with x on the left-hand side: (7x - 3x) - 2 = 4x This simplifies to: 4x - 2 = 4x
02

Isolate the Variable Term

Move all terms involving x to one side of the equation. Currently, we have: 4x - 2 = 4x Subtract 4x from both sides to isolate the variable: 4x - 4x - 2 = 4x - 4x This simplifies to: -2 = 0
03

Analyze the Result

The result from Step 2 is -2 = 0 which is not a true statement. This means that there is no value of x that can satisfy the original equation.

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Key Concepts

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

Combining Like Terms
Combining like terms is an essential step in solving linear equations. It involves simplifying the equation by merging terms that have the same variable. For example, in the equation provided:
\[ 7x - 2 - 3x = 4x \]
We see two terms on the left with the variable x, which are 7x and -3x. By combining these, we get:
\[ (7x - 3x) - 2 = 4x \]
This simplifies to: \[ 4x - 2 = 4x \]
Combining like terms helps in reducing the complexity of the equation and makes it easier to isolate the variable.
Isolating the Variable
Isolating the variable is the process of manipulating an equation so that the variable is alone on one side of the equation. In our example:
\[ 4x - 2 = 4x \]
We need to isolate x. To do this, we subtract 4x from both sides:
\[ 4x - 4x - 2 = 4x - 4x \]
This simplifies to: \[ -2 = 0 \]
By isolating the variable, we aim to find its value. However, here it led us to an inconsistency, which means further analysis is needed.
No Solution in Algebra
Sometimes, during the process of solving an equation, we might end up with a statement that is always false, such as \[ -2 = 0 \]. This indicates that there is no value of the variable that will satisfy the equation. In algebra, such a situation is known as 'no solution.'
When we end up with a contradiction instead of a valid result, like in our case:
\[ -2 = 0 \]
It means the equation has no solution. This outcome is a critical aspect of understanding the types of solutions one can encounter while solving linear equations.

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Most popular questions from this chapter

Calculate using the rules for order of operations. Use the commutative, associative, and distributive laws to show that \(5(a+b c)\) is equivalent to \(c(b \cdot 5)+a \cdot 5 .\) Use only one law in each step of your work.

Solve and check. The symbol \(\sqsubseteq\) indicates an exercise designed to be solved with a calculator. $$ 6 x-\\{5 x-[7 x-(4 x-(3 x+1))]\\}=6 x+5 $$

Write an equivalent expression using the distributive law. $$ 3(a+5) $$

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ \frac{3^{4}-(5-3)^{4}}{8-2^{3}} $$

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ 2^{2}-5^{2} $$
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