Textbook Question
In Exercises 1–34, solve each rational equation. If an equation has no solution, so state.(x−6)/(x+5) = (x−3)/(x+1)
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1. Equations & Inequalities
Linear Equations
2:09 minutesProblem 11
Textbook Question
Solve and check each linear equation. 2x - 7 = 6 + x
Verified step by step guidance1
Start with the given linear equation: \$2x - 7 = 6 + x$.
Isolate the variable terms on one side by subtracting \(x\) from both sides: \$2x - x - 7 = 6 + x - x\(, which simplifies to \)x - 7 = 6$.
Next, isolate the variable term by adding 7 to both sides: \(x - 7 + 7 = 6 + 7\), which simplifies to \(x = 13\).
The solution to the equation is \(x = 13\).
Check the solution by substituting \(x = 13\) back into the original equation: verify that \$2(13) - 7\( equals \)6 + 13$.
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2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Linear Equations
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. It forms a straight line when graphed and typically has the form ax + b = c. Understanding how to manipulate and solve these equations is fundamental in algebra.
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Categorizing Linear Equations
Isolating the Variable
Isolating the variable means rearranging the equation to get the variable alone on one side. This involves using inverse operations such as addition, subtraction, multiplication, or division to simplify the equation step-by-step until the variable's value is found.
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Checking the Solution
After solving the equation, substituting the solution back into the original equation verifies its correctness. This step ensures that the value satisfies the equation and helps identify any errors made during the solving process.
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