QUESTION IMAGE
Question
given: $ce = cd + de$; $cd = (7x - 5)$; $de = (2x + 5)$; $ce = (x + 8)$
prove: $x = 1$
diagram: segment $ce$ with $c$, $d$, $e$; $cd = 7x - 5$, $de = 2x + 5$, $ce = x + 8$
statements\t\t\t\treasons
- $ce = cd + de$; $cd = (7x - 5)$; $de = (2x + 5)$; $ce = (x + 8)$\t1. given
- $x + 8 = (7x - 5) + (2x + 5)$\t\t\t2. blank
- $x + 8 = 9x$\t\t\t\t3. simplify
- blank\t\t\t\t4. subtraction property of equality
- $1 = x$\t\t\t\t5. blank
- $x = 1$\t\t\t\t6. blank
a. reflexive property of equality
b. $x = 9x - 8$
c. division property of equality
d. symmetric property of equality
e. $8 = 8x$
f. substitution property of equality
Step 1: Analyze Statement 2 Reason
To get \( x + 8=(7x - 5)+(2x + 5) \) from the given \( CE = CD + DE \), \( CD=(7x - 5) \), \( DE=(2x + 5) \), \( CE=(x + 8) \), we substitute the values of \( CE \), \( CD \), and \( DE \) into the equation \( CE = CD + DE \). This is the Substitution Property of Equality (option f).
Step 2: Analyze Statement 4
Using the Subtraction Property of Equality (subtract \( x \) from both sides of \( x + 8 = 9x \)), we get \( 8=9x - x \) which simplifies to \( 8 = 8x \)? Wait, no, wait. Wait, if we use Subtraction Property of Equality on \( x + 8=9x \), subtract \( x \) from both sides: \( x + 8-x=9x - x \), so \( 8 = 8x \)? Wait, no, the options have \( e. 8 = 8x \) and \( b. x=9x - 8 \). Wait, maybe I made a mistake. Wait, the Subtraction Property of Equality: if \( a = b \), then \( a - c=b - c \). So starting from \( x + 8=9x \), subtract \( x \) from both sides: \( x + 8-x=9x - x \) → \( 8 = 8x \) (option e)? Wait, no, the options for statement 4: the statement is what we get after applying Subtraction Property of Equality. Wait, the equation is \( x + 8=9x \). Subtract \( x \) from both sides: \( x + 8 - x=9x - x \) → \( 8 = 8x \) (option e). Wait, but let's check the options again. The options for statement 4 are the possible statements. The reason is Subtraction Property of Equality, so we do an operation on both sides. Let's re-express:
Given \( x + 8=9x \), subtract \( x \) from both sides: \( x + 8 - x=9x - x \) → \( 8 = 8x \) (option e). Wait, but the options also have \( b. x = 9x - 8 \), which would be subtracting 8 from both sides: \( x + 8-8=9x - 8 \) → \( x=9x - 8 \). Oh, I see, I messed up the operation. The Subtraction Property of Equality can be subtracting a term from both sides. If we subtract 8 from both sides: \( x + 8-8=9x - 8 \) → \( x=9x - 8 \) (option b). Wait, now I'm confused. Let's check the steps again.
Step 2: Substitution (f) to get \( x + 8=(7x - 5)+(2x + 5) \)
Step 3: Simplify \( (7x - 5)+(2x + 5)=9x \), so \( x + 8=9x \)
Step 4: Reason is Subtraction Property of Equality. So we need to perform a subtraction on both sides. Let's see the options for statement 4:
Option b: \( x=9x - 8 \) → obtained by subtracting 8 from both sides of \( x + 8=9x \) (since \( x + 8-8=9x - 8 \) → \( x=9x - 8 \))
Option e: \( 8 = 8x \) → obtained by subtracting \( x \) from both sides of \( x + 8=9x \) ( \( x + 8-x=9x - x \) → \( 8 = 8x \))
Now, let's see the next step: step 5 is \( 1 = x \), which would come from dividing both sides by 8 (if step 4 is \( 8 = 8x \), then divide by 8: \( 1 = x \), which is option c (Division Property of Equality) for step 5. Then step 6 is \( x = 1 \), which is Symmetric Property of Equality (option d), since if \( 1 = x \), then \( x = 1 \).
Let's re-express the steps correctly:
- Given (as is)
- Substitution Property (f): substitute \( CE = x + 8 \), \( CD = 7x - 5 \), \( DE = 2x + 5 \) into \( CE = CD + DE \), so \( x + 8=(7x - 5)+(2x + 5) \)
- Simplify: \( (7x - 5)+(2x + 5)=9x \), so \( x + 8=9x \)
- Subtraction Property of Equality: subtract \( x \) from both sides: \( x + 8 - x=9x - x \) → \( 8 = 8x \) (option e)
- Division Property of Equality (c): divide both sides by 8: \( \frac{8}{8}=\frac{8x}{8} \) → \( 1 = x \)
- Symmetric Property of Equality (d): if \( 1 = x \), then \( x = 1 \)
Final Answers for Each Blank:
- Statement 2 Reason: f. Substitution Property of Equality
- Statement 4: e. \( 8 = 8x \)
- Statement 5 Reason: c. Division Property of Equality
- Statement 6 Reason: d. Symmetric Property of Equality
(Assuming th…
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Step 1: Analyze Statement 2 Reason
To get \( x + 8=(7x - 5)+(2x + 5) \) from the given \( CE = CD + DE \), \( CD=(7x - 5) \), \( DE=(2x + 5) \), \( CE=(x + 8) \), we substitute the values of \( CE \), \( CD \), and \( DE \) into the equation \( CE = CD + DE \). This is the Substitution Property of Equality (option f).
Step 2: Analyze Statement 4
Using the Subtraction Property of Equality (subtract \( x \) from both sides of \( x + 8 = 9x \)), we get \( 8=9x - x \) which simplifies to \( 8 = 8x \)? Wait, no, wait. Wait, if we use Subtraction Property of Equality on \( x + 8=9x \), subtract \( x \) from both sides: \( x + 8-x=9x - x \), so \( 8 = 8x \)? Wait, no, the options have \( e. 8 = 8x \) and \( b. x=9x - 8 \). Wait, maybe I made a mistake. Wait, the Subtraction Property of Equality: if \( a = b \), then \( a - c=b - c \). So starting from \( x + 8=9x \), subtract \( x \) from both sides: \( x + 8-x=9x - x \) → \( 8 = 8x \) (option e)? Wait, no, the options for statement 4: the statement is what we get after applying Subtraction Property of Equality. Wait, the equation is \( x + 8=9x \). Subtract \( x \) from both sides: \( x + 8 - x=9x - x \) → \( 8 = 8x \) (option e). Wait, but let's check the options again. The options for statement 4 are the possible statements. The reason is Subtraction Property of Equality, so we do an operation on both sides. Let's re-express:
Given \( x + 8=9x \), subtract \( x \) from both sides: \( x + 8 - x=9x - x \) → \( 8 = 8x \) (option e). Wait, but the options also have \( b. x = 9x - 8 \), which would be subtracting 8 from both sides: \( x + 8-8=9x - 8 \) → \( x=9x - 8 \). Oh, I see, I messed up the operation. The Subtraction Property of Equality can be subtracting a term from both sides. If we subtract 8 from both sides: \( x + 8-8=9x - 8 \) → \( x=9x - 8 \) (option b). Wait, now I'm confused. Let's check the steps again.
Step 2: Substitution (f) to get \( x + 8=(7x - 5)+(2x + 5) \)
Step 3: Simplify \( (7x - 5)+(2x + 5)=9x \), so \( x + 8=9x \)
Step 4: Reason is Subtraction Property of Equality. So we need to perform a subtraction on both sides. Let's see the options for statement 4:
Option b: \( x=9x - 8 \) → obtained by subtracting 8 from both sides of \( x + 8=9x \) (since \( x + 8-8=9x - 8 \) → \( x=9x - 8 \))
Option e: \( 8 = 8x \) → obtained by subtracting \( x \) from both sides of \( x + 8=9x \) ( \( x + 8-x=9x - x \) → \( 8 = 8x \))
Now, let's see the next step: step 5 is \( 1 = x \), which would come from dividing both sides by 8 (if step 4 is \( 8 = 8x \), then divide by 8: \( 1 = x \), which is option c (Division Property of Equality) for step 5. Then step 6 is \( x = 1 \), which is Symmetric Property of Equality (option d), since if \( 1 = x \), then \( x = 1 \).
Let's re-express the steps correctly:
- Given (as is)
- Substitution Property (f): substitute \( CE = x + 8 \), \( CD = 7x - 5 \), \( DE = 2x + 5 \) into \( CE = CD + DE \), so \( x + 8=(7x - 5)+(2x + 5) \)
- Simplify: \( (7x - 5)+(2x + 5)=9x \), so \( x + 8=9x \)
- Subtraction Property of Equality: subtract \( x \) from both sides: \( x + 8 - x=9x - x \) → \( 8 = 8x \) (option e)
- Division Property of Equality (c): divide both sides by 8: \( \frac{8}{8}=\frac{8x}{8} \) → \( 1 = x \)
- Symmetric Property of Equality (d): if \( 1 = x \), then \( x = 1 \)
Final Answers for Each Blank:
- Statement 2 Reason: f. Substitution Property of Equality
- Statement 4: e. \( 8 = 8x \)
- Statement 5 Reason: c. Division Property of Equality
- Statement 6 Reason: d. Symmetric Property of Equality
(Assuming the question is to fill in the blanks for the reasons and statement 4, here are the answers for each blank as per the analysis above.)