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Chapter 2: Problem 48

Solve the equations. More than one step is needed in each case.$$-4-6 x=44$$

Short Answer

Expert verified
The solution is \(x = -8\).

Step by step solution

01

Isolate the variable term

Start by eliminating the constant term from the left side of the equation. Add 4 to both sides to keep the equation balanced:\[-4 - 6x + 4 = 44 + 4\]This simplifies to:\[-6x = 48\]
02

Solve for the variable

Divide both sides of the equation by -6 to solve for \(x\):\[x = \frac{48}{-6}\]This simplifies to:\[x = -8\]

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Key Concepts

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

Step-by-Step Solution
When dealing with equations that require several operations to solve, following a detailed step-by-step method can be very effective. This means breaking down the problem into smaller, simpler tasks that are easier to manage. Starting with the equation -4 - 6x = 44, we notice that it involves both numbers (constants) and variables (terms with x). By approaching it step by step, we make sure that every action we take is purposeful and leads us closer to the solution.
  • **Identify the Variable and Constants:** Spot where the variable is (here, it's -6x), and note the constants ( -4 and 44).
  • **Perform One Operation at a Time:** Deal with one part of the equation first. Our first task is to remove the constant on the same side as the variable.
  • **Check Each Step:** As you complete each operation, check your work. Verify that each new equation produced still holds true. This avoids unnecessary rework later.
By sticking to a sequential process, we not only find the solution, but also understand the logical flow of algebraic procedures.
Isolate the Variable
Isolating the variable is a crucial step in solving equations like -4 - 6x = 44. The basic idea is to rearrange the equation so that the variable stands alone on one side. This helps in finding the precise value of the variable. To begin, let's focus on getting -6x by itself. On the left side, we have -4 as a hindering constant. To eliminate it, we add 4 to both sides of the equation. This is done because subtracting and then adding the same number essentially cancels it out.
  • Plus 4 on left side: -4 + 4 becomes 0, leaving us with just -6x.
  • Plus 4 on right side: 44 + 4 turns the equation into 48. Now it reads -6x = 48.
By performing these operations, we've successfully isolated the variable term. This paves the way for the next steps in finding what the variable actually equals.
Balancing Equations
Balancing equations is like walking on a tightrope 鈥� you must remain level and stable on both sides. In equation-solving, whatever change is made to one side must also be made to the other. This keeps the equation valid and true. In the equation -4 - 6x = 44, when we added 4 to both sides, we performed a balancing act. We were ensuring the equation remains equal by maintaining symmetry:
  • Addition of 4: Both left and right sides change, nullifying the constant -4 while increasing the numeric value on the right.
  • Division for Solving: To solve for the isolated variable x, we divide each side by -6, again ensuring both sides receive the same treatment. -6x/ -6 = 48/ -6 leads us to x = -8, maintaining the balance of the equation.
These balance-maintaining actions make sure our solution isn't just a number, but the right number. It's the core of solving equations effectively.

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Most popular questions from this chapter

Solve the equations. More than one step is needed in each case.$$4 x+3=31$$

Simplify the expressions, which involve exponents and square roots. Round the results to two decimal places as necessary. $$\frac{2.026}{\sqrt{17}}$$

Classify each of the inequalities as true or false. $$-7.2 E-2 \leq 0.072$$

Consider the formula $$\frac{0.55-p}{\sqrt{\frac{p(1-p)}{n}}}$$ Evaluate this expression using the given values. Round the results to three decimal places. $$p=0.5, n=44$$

Consider a variable \(x\) where \(x\) represents the whole numbers from 0 to 20. Stated mathematically, the possible values of \(x\) are $$x=0,1,2, \ldots, 19,20$$ For each of the phrases, write an inequality and then determine which values satisfy that inequality. \(x\) is more than 15
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