Problem 40 Solve. $$ 7|z|+2=16 $$... [FREE SOLUTION]

Chapter 8: Problem 40

Solve. $$ 7|z|+2=16 $$

Short Answer

Expert verified

The solutions are $ z = 2 $ and $ z = -2 $.

Step by step solution

Isolate the Absolute Value

Subtract 2 from both sides of the equation to isolate the term with the absolute value. You get: $ 7|z| + 2 - 2 = 16 - 2 $This simplifies to: $ 7|z| = 14 $

Solve for the Absolute Value

Divide both sides of the equation by 7 to solve for the absolute value. You get: $ |z| = \frac{14}{7} $This simplifies to: $ |z| = 2 $

Set Up Two Equations

Since the absolute value of $ z $ is 2, set up the two possible equations: $ z = 2 $ and $ z = -2 $

Solve Each Equation

The solutions to the equations are: $ z = 2 $ and $ z = -2 $

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Key Concepts

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

Isolate the Absolute Value

When solving absolute value equations, the first step is often to isolate the absolute value expression itself. This makes the equation easier to handle and simplifies subsequent steps.

For example, in the equation $7|z| + 2 = 16$, we start by removing any constants on the same side as the absolute value term. Here, subtract 2 from both sides to get:
\[ 7|z| + 2 - 2 = 16 - 2 \] This simplifies to:
\[ 7|z| = 14 \] This setup ensures that we only have the absolute value term on one side of the equation, isolated and ready for the next step.

Setting Up Two Equations

After isolating the absolute value, it鈥檚 essential to remember that absolute values can be positive or negative because they measure distance from zero. Hence, the equation inside the absolute value can be equal to the positive or negative of the number on the other side of the equation.

Using the ongoing example, once we have \[ 7|z| = 14 \] we need to solve for $|z|$ by dividing both sides by 7, giving us:
\[ |z| = \frac{14}{7} \] Simplifying, we find:
\[ |z| = 2 \]
Now, set up two separate linear equations to reflect the definition of absolute value:

$z = 2$
$z = -2$

These equations represent both scenarios that satisfy the absolute value condition.

Solving Linear Equations

Finally, we solve the linear equations obtained from the previous step. Each equation represents a potential solution for the original absolute value equation.

For $z = 2$, the solution is straightforward since no further steps are needed. $z = 2$.
Similarly, for $z = -2$, the solution is just as direct: $z = -2$.

Together, these solutions cover all possible values of $z$ that satisfy the original equation.

This is a vital part of solving absolute value equations. Linear equations are simpler and more straightforward, but always remember to verify solutions in the context of the original equation to avoid extraneous solutions.

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91影视

Solve. $$ 7|z|+2=16 $$

Short Answer

Step by step solution

Isolate the Absolute Value

Solve for the Absolute Value

Set Up Two Equations

Solve Each Equation

Key Concepts

Isolate the Absolute Value

Setting Up Two Equations

Solving Linear Equations

One App. One Place for Learning.

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