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Chapter 1: Problem 34

Solve each equation, if possible. $$ 1-\frac{1}{2} x=6 $$

Short Answer

Expert verified
The solution is $$ x = -10 $$.

Step by step solution

01

Isolate the Variable

First, transform the equation to isolate the term containing the variable. Subtract 1 from both sides of the equation:yielding$$1 - \frac{1}{2} x - 1 = 6 - 1$$This simplifies to: $$-\frac{1}{2} x = 5$$
02

Eliminate the Coefficient

To remove the fraction, multiply both sides of the equation by \the reciprocal of $$-\frac{1}{2}$$ which is -2. So, multiplying both sides by -2 gives:$$-2 \times -\frac{1}{2} x = 5 \times -2$$This simplifies to:$$ x = -10 $$

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Key Concepts

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

isolating variables
When solving linear equations, your first goal is often to isolate the variable you want to solve for. This means you want to get the term with the variable alone on one side of the equation. Let's look at an example: if you have the equation \(1 - \frac{1}{2} x = 6\), you need to start by eliminating any constants on the same side as the variable. In this case, you subtract 1 from both sides, transforming the equation to \(1 - \frac{1}{2} x - 1 = 6 - 1\). This simplifies to \(-\frac{1}{2} x = 5\). Breaking each step down into manageable parts can make it much easier to see what you need to do next.
eliminating coefficients
After isolating the variable, the next step is to eliminate any coefficients multiplying the variable. Coefficients are numbers that are multiplied by variables. In our example, the coefficient of \(x\) is \(-\frac{1}{2}\). To remove this coefficient, you multiply both sides of the equation by its reciprocal. A reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\). So, the reciprocal of \(-\frac{1}{2}\) is \(-2\). Multiplying both sides by \(-2\) gives us:
  • \(-2 \times -\frac{1}{2} x = 5 \times -2\)
, which simplifies to \(x = -10\). Now, the variable is fully isolated, and the solution is found.
fractions in equations
Fractions can make equations look complex, but don鈥檛 be intimidated! Remember that you can eliminate them by using their reciprocals. In our equation \(1 - \frac{1}{2} x = 6\), the fraction \(\frac{1}{2}\) is part of the coefficient of \(x\). You reduced this by multiplying the whole equation by \(-2\), which is the reciprocal. This conversion simplifies the fraction, making the equation much easier to work with. Always aim to clear fractions early in the solving process to simplify your work.
basic algebra
Basic algebra involves understanding and manipulating mathematical symbols and expressions to find unknown values. Key principles include operations like addition, subtraction, multiplication, and division, which are used conjunctively to isolate variables and solve equations. For instance, in the given equation, starting with \(1 - \frac{1}{2} x = 6\), you used subtraction to isolate the variable-containing term and then multiplication to remove the coefficient. Practicing these basic operations builds a strong foundation for tackling more complex problems in algebra and beyond.

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Most popular questions from this chapter

Computing Grades In your Economics 101 class, you have scores of \(68,82,87,\) and 89 on the first four of five tests. To get a grade of B, the average of the first five test scores must be greater than or equal to 80 and less than 90 . (a) Solve an inequality to find the least score you can get on the last test and still earn a \(\mathrm{B}\). (b) What score do you need if the fifth test counts double?

Find \(k\) so that the equation \(x^{2}-k x+4=0\) has a repeated real solution.

Find the real solutions of each equation by factoring. $$ 3 x\left(x^{2}+2 x\right)^{1 / 2}-2\left(x^{2}+2 x\right)^{3 / 2}=0 $$

Find the real solutions, if any, of each equation. Use any method. $$ \frac{1}{2} x^{2}=\sqrt{2} x+1 $$

Challenge Problem Show that the real solutions of the equation \(a x^{2}+b x+c=0, a \neq 0,\) are the negatives of the real solutions of the equation \(a x^{2}-b x+c=0\). Assume that \(b^{2}-4 a c \geq 0\)
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