write a two column proof given 7y=8x-14 y=6
prove: x=7
Statements: Reasons:
1. 7y = 8x - 14 1. Given
2. 7(6) = 8x - 14 2. Substituted y with 6 (from y = 6)
3. 42 = 8x - 14 3. Simplified
4. 42 + 14 = 8x 4. Added 14 to both sides
5. 56 = 8x 5. Simplified
6. x = 7 6. Divided both sides by 8
Given: 7y = 8x - 14 and y = 6
To prove: x = 7
Proof:
Statement | Reason
__________________________________|____________________
1. y = 6 | Given
2. 7(6) = 8x - 14 | Substitute y = 6 into 7y = 8x - 14
3. 42 = 8x - 14 | Simplify 7(6) and combine like terms
4. 42 + 14 = 8x | Add 14 to both sides of the equation
5. 56 = 8x | Simplify 42 + 14
6. (56/8) = (8x/8) | Divide both sides of the equation by 8
7. 7 = x | Simplify (56/8) and (8x/8)
8. x = 7 | Rearrange the equation to have x on the left side
Therefore, the two-column proof shows that if 7y = 8x - 14 and y = 6, then x = 7.
To write a two-column proof, we need to list the statements and the reasons for each step. Here's a proof for the given statement:
Statement | Reason
--- | ---
1. 7y = 8x - 14 | Given
2. y = 6 | Given
3. 7(6) = 8x - 14 | Substituting 'y' with '6' in Statement 1 (Substitution property of equality)
4. 42 = 8x - 14 | Performing the multiplication in Statement 3
5. 42 + 14 = 8x | Adding 14 to both sides to isolate '8x' (Addition property of equality)
6. 56 = 8x | Simplifying Statement 5
7. 56/8 = x | Dividing both sides by 8 to solve for 'x' (Division property of equality)
8. 7 = x | Simplifying Statement 7
Therefore, we have proved that x = 7.