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Chapter 6: Problem 3

Solve and check each equation. \(x+5=-12\)

Short Answer

Expert verified
The solution to the equation is \(x = -17\).

Step by step solution

01

Isolate the variable

Isolate 'x' on one side of the equation by adding -5 to both sides, as follows: \(x + 5 - 5 = -12 -5\). This simplifies to \(x = -17\)
02

Check the solution

Insert the found value of 'x' back in the original equation to verify the solution: \(x + 5 = -17 + 5\), which yields \(-12 = -12\). Both sides are equal, so the solution is correct.

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Key Concepts

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

Isolate Variable
The process of solving linear equations often starts with the goal of isolating the variable. This simply means to get the variable by itself on one side of the equation. To achieve this, balance is key 鈥� whatever you do to one side of the equation, you must do to the other.

Let's look at our example, where the initial equation is \(x+5=-12\). Our objective is to find the value of \(x\).
  • Firstly, we notice that \(x\) is being increased by 5. So, to isolate \(x\), we need to eliminate that +5.
  • We do this by performing the inverse operation, which in this case, is subtraction. By subtracting 5 from both sides, we stay balanced and get \(x+5-5=-12-5\), simplifying to \(x=-17\).
Remember to keep operations consistent across both sides, as symmetry is what keeps our equation valid.
Equation Checking
After isolating the variable and arriving at a solution, it's always wise to perform equation checking. This is our way of verifying that the solution we found is valid. We do this by substituting our solution back into the original equation to see if it holds true.

In our example, we found that \(x=-17\). To check:
  • We place -17 back into the original equation: \((-17) + 5 = -12\).
  • When simplifying this, we do indeed get \(-12 = -12\), indicating that our left and right sides match, and thus confirming that our solution is correct.
This step shouldn't be skipped as it ensures no mistakes were made along the way and strengthens your understanding and confidence in solving equations.
Algebraic Thinking
Behind every equation solving process lies algebraic thinking. It's a form of reasoning that allows us to understand relationships and solve problems involving unknown values. This kind of thinking isn't just about performing calculations; it's about understanding the 'why' and the 'how'.

Let's dig deeper into our given problem. Algebraic thinking involves recognizing patterns and structures:
  • We know that addition and subtraction are inverse operations, so they undo each other. We apply this knowledge to move terms from one side of the equation to the other.
  • We also observe the balance of an equation akin to a scale, maintaining equality despite the manipulations that isolate the variable.
  • Finally, checking our work through substitution helps reinforce our comprehension and uphold the integrity of algebraic principles.
This type of thinking is fundamental in not just solving equations, but in all of mathematics and its applications.

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Most popular questions from this chapter

Use FOIL to find the products in Exercises 1-8. \((2 x-5)(x+3)\)

Solve each equation using the zero-product principle. \((x+11)(x-5)=0\)

Solve the equations in Exercises 53-72 using the quadratic formula. \(3 x^{2}=5 x-1\)

Solve the quadratic equations in Exercises 37-52 by factoring. \(x^{2}-4 x=21\)

Solve the equations in Exercises 53-72 using the quadratic formula. \(x^{2}+8 x+12=0\)
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