Complete the missing reasons for the proof.
Given: [tex]4(x-2)=6 x+18[/tex]
Prove: [tex]x=-13[/tex]
| Statements | Reasons |
|---|---|
| 1. [tex]4(x-2)=6 x+18[/tex] | Given |
| 2. [tex]4 x-8=6 x+18[/tex] | Distributive property |
| 3. [tex]-2 x-8=18[/tex] | |
| 4. [tex]-2 x=26[/tex] | Addition property of equality |
| 5. [tex]x=-13[/tex] | |
Which of the following correctly fills in the missing reasons?
A. addition property of equality; division property of equality
B. subtraction property of equality; multiplication property of equality
C. subtraction property of equality; division property of equality
D. addition property of equality; multiplication property of equality
Answer (2)
The missing reasons in the proof are: Step 3 uses the Subtraction Property of Equality, and Step 5 uses the Division Property of Equality. This completes the logical flow from the original equation to the concluded value of x = − 13 .
;
Apply the distributive property to get 4 x − 8 = 6 x + 18 .
Subtract 6 x from both sides: − 2 x − 8 = 18 .
Add 8 to both sides: − 2 x = 26 .
Divide by − 2 : x = − 13 , so the final answer is x = − 13 .
Explanation
Analyzing the Problem We are given the equation 4 ( x − 2 ) = 6 x + 18 and we want to prove that x = − 13 . We need to fill in the missing reasons in the proof table. Let's analyze the steps.
Steps 1 and 2 Step 1: 4 ( x − 2 ) = 6 x + 18 is given. Step 2: 4 x − 8 = 6 x + 18 is obtained by applying the distributive property to the left side of the equation in Step 1: 4 ( x − 2 ) = 4 x − 4 ( 2 ) = 4 x − 8 .
Step 3: Subtraction Property of Equality Step 3: − 2 x − 8 = 18 is obtained from 4 x − 8 = 6 x + 18 . To get from 4 x − 8 to − 2 x − 8 , we need to subtract 6 x from both sides of the equation: 4 x − 8 − 6 x = 6 x + 18 − 6 x , which simplifies to − 2 x − 8 = 18 . Therefore, the reason for Step 3 is the subtraction property of equality.
Step 4: Addition Property of Equality Step 4: − 2 x = 26 is obtained from − 2 x − 8 = 18 . To get from − 2 x − 8 to − 2 x , we need to add 8 to both sides of the equation: − 2 x − 8 + 8 = 18 + 8 , which simplifies to − 2 x = 26 . Therefore, the reason for Step 4 is the addition property of equality.
Step 5: Division Property of Equality Step 5: x = − 13 is obtained from − 2 x = 26 . To get from − 2 x to x , we need to divide both sides of the equation by − 2 : − 2 − 2 x = − 2 26 , which simplifies to x = − 13 . Therefore, the reason for Step 5 is the division property of equality.
Conclusion Therefore, the missing reasons are: Step 3: subtraction property of equality; Step 5: division property of equality.
Examples
Imagine you're balancing a budget where expenses on both sides must remain equal. Using the properties of equality, like addition, subtraction, multiplication, and division, ensures that if you adjust one side, you adjust the other accordingly to maintain balance. These algebraic principles are fundamental in managing finances, calculating costs, and ensuring financial stability.
