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Chapter 1: Problem 48

Solve each of the given equations for \(\mathrm{x}\). $$-48 x+95=0$$

Short Answer

Expert verified
\(x = \frac{95}{48}\)

Step by step solution

01

Identify Equation

The equation given is \(-48x + 95 = 0\). We need to solve for \(x\).
02

Isolate the Variable

To isolate \(x\), we first need to eliminate the constant \(+95\) from the left side of the equation. We do this by subtracting \(95\) from both sides:\[-48x + 95 - 95 = 0 - 95\] This simplifies to:\(-48x = -95\).
03

Solve for x

To solve for \(x\), we need to get \(x\) by itself. We do this by dividing both sides of the equation by \(-48\):\[x = \frac{-95}{-48}\] This simplifies further to: \[x = \frac{95}{48}\].

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Key Concepts

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

Equation Isolation
Equation isolation is a key step when solving linear equations. The objective is to have the variable alone on one side of the equation. This makes it easier to identify its value. To start, observe what is on the same side as the variable. Examine whether any constants or other terms are present. For example, in the equation \(-48x + 95 = 0\), our goal is to isolate \(x\). Notice the constant \(+95\) with \(x\). To remove it, perform the opposite operation. In this case, subtract \(95\) from both sides. It results in simplifying to \(-48x = -95\). Key points to remember:
  • The rule of performing the same operation on both sides maintains the balance of the equation.
  • Identify operations that keep the variable isolated.
By following these steps, you can effectively simplify the equation.
Solving for Variables
With the variable isolated, the next step is solving for its value. This involves transforming the equation into a form where the variable stands alone. Let's take a deeper look at how this works with \(-48x = -95\).Division is often used when you are solving for the variable. To get \(x\) by itself:- Divide both sides by the coefficient of \(x\), which in this case is \(-48\).Thus, the division yields \(x = \frac{-95}{-48}\). This division step is crucial as it allows us to express \(x\) as a fraction.A few important tips:
  • Understand dividing or multiplying on both sides of the equation cancels out the coefficient.
  • Maintain accuracy when calculating fractions or decimal results.
This approach clarifies finding the variable's value and assures correctness in solutions.
Algebraic Simplification
Algebraic simplification is about making an expression as straightforward as possible. After isolating \(x\) and computing \(x = \frac{-95}{-48}\), the task shifts to simplifying this fraction.In simplification, both the numerator and the denominator can often be divided by their greatest common divisor. However, in the result \(x = \frac{95}{48}\), since both numbers do not have a common divisor, the expression \(\frac{95}{48}\) is already in its simplest form.Consider these benefits of algebraic simplification:
  • Simplified expressions are easier to interpret and work with.
  • It can reveal further relationships between variables when dealing with complex equations.
  • Reduces potential errors in further calculations.
Simplifying algebraic expressions enhances clarity and efficiency, especially in longer calculations.

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Most popular questions from this chapter

Solve the inequality. Express your answer in both interval and set notations, and shade the solution on a number line. $$6 x+3<8 x+8$$

Solve the compound inequality. Express your answer in both interval and set notations, and shade the solution on a number line. \(9 x-9 \leq 9\) and \(5 x>-1\)

Solve each of the given equations for \(\mathrm{x}\). $$(8 x-3)+(-3 x+9)=-4 x-7$$

Solve each of the given equations for \(\mathrm{x}\). $$(7 x-9)-(9 x+4)=-3 x+2$$

Solve each of the given equations for \(\mathrm{x}\). $$-7 x+4=-6$$
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