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Chapter 3: Problem 20

Solve the equation and check your solution. $$7 x-21=0$$

Short Answer

Expert verified
The solution of the equation is \( x = 3 \)

Step by step solution

01

Rewrite the equation

The given equation is \( 7x - 21 = 0 \). To isolate the variable \( x \), we need to move the constant to the other side. This can be done by adding 21 to both sides of the equation according to the equality property. This gives us the new equation: \( 7x = 21 \).
02

Solve for x

Now, to isolate \( x \), we need to eliminate the coefficient of \( x \). This is done by dividing both sides of the equation by 7. This yields the solution \( x = 3 \).
03

Check the obtained solution

We can verify the obtained solution by substituting the value of \( x \) in the original equation. Substituting \( x = 3 \) in the equation \( 7x - 21 = 0 \) we obtain \( 21 - 21 = 0 \) which is true, indicating that the solution \( x = 3 \) is correct.

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Key Concepts

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

Isolate the Variable
Solving linear equations often involves the crucial step of isolating the variable. To 'isolate' simply means to get the variable by itself on one side of the equation. Take the equation from the exercise, 7x - 21 = 0. The goal here is to maneuver the equation to have x alone on one side.

How do we do this? By performing the reverse operations. Since 7x implies that x is multiplied by 7, and -21 is subtracted from 7x, we do the opposite to isolate x; we add 21 to both sides to eliminate the -21, and then divide both sides by 7 to counter the multiplication. This results in x = 3. It's all about reversing the math that's been done to the variable to get it by itself.
Equality Property
The equality property is a foundational rule that allows us to maintain balance in an equation. It states that if we perform the same operation on both sides of an equation, the equation remains balanced or equal. This is key when manipulating equations to solve for a variable.

In the given exercise, we used the equality property when we added 21 to both sides of 7x - 21 = 0, resulting in 7x = 21. When isolating the variable, it's vital to remember that whatever is done to one side must be done to the other to maintain the equality. This is what upholds the truth of the equation through our operations and ultimately leads us to the correct solution.
Check the Solution
After finding the solution to an equation, it's important not to assume the work is done. Always 'check the solution' to ensure accuracy. This step verifies that the value obtained actually satisfies the original equation.

To check the solution from our exercise, we substitute x = 3 back into the original equation, resulting in 7(3) - 21. Simplifying this we get 21 - 21, which equals 0. This matches the original equation's right side, confirming our solution is correct. Effectively, checking the solution is like running a trial to make sure the 'answer' isn't just another 'guess'. It's a good habit to catch mistakes and bolster confidence in the result.

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Most popular questions from this chapter

Match the statement with the property it represents. (a) Addition Property of Inequality (b) Subtraction Property of Inequality (c) Multiplication Property of Inequality (d) Division Property of Inequality \(-8>-9\), so \(-8+4>-9+4\).

Write a verbal description of the inequality and sketch its graph. $$-3 \geq t>-3.8$$

Determine whether each value of \(x\) is a solution of the inequality. \(\frac{2}{3} x+4<6\) (a) \(x=7\) (b) \(x=0\) (c) \(x=-\frac{1}{2}\) (d) \(x=3\)

Determine whether the statement is true or false. Justify your answer. The inequality \(-\frac{1}{2} x+6>0\) is equivalent to \(x>12\).

Solve the proportion. $$\frac{34}{x}=\frac{102}{48}$$
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