Write a direct algebraic proof to show that if
a/b=c/d, then a+c=b+d
To prove the statement "if a/b = c/d, then a + c = b + d" directly using algebra, we need to start with the assumption that a/b = c/d and then use algebraic properties and manipulations to arrive at the conclusion that a + c = b + d.
Here's the direct algebraic proof:
Given: a/b = c/d
Step 1: Start with the assumption that a/b = c/d.
Step 2: Cross-multiply to eliminate the fractions. Multiply a by d and c by b.
(ad)/(bd) = (cb)/(bd)
Step 3: Since ad/bd = a/b and cb/bd = c/d, we can substitute the original equal fractions.
a/b = c/d
Step 4: Use the transitive property of equality to combine the two equations.
(ad)/(bd) = (cb)/(bd) = a/b = c/d
Step 5: Cancel out the common factor (bd) on both sides.
ad = cb
Step 6: Re-arrange the equation to group like terms together.
ad - cb = 0
Step 7: Add (bd) to both sides of the equation.
ad - cb + bd = 0 + bd
Step 8: Factor out (d - b) on the left side.
(d - b)(a) = bd
Step 9: Divide both sides by (d - b).
a = bd / (d - b)
Step 10: Distribute (a) on the right side.
a = (bd/d) - (bd/b)
a = b - c
Step 11: Add (c) to both sides of the equation.
a + c = b
Step 12: Using the commutative property of addition, we can rewrite the equation as:
b + a = b + c
Step 13: Finally, by using the transitive property of equality, we obtain:
a + c = b + d
Therefore, we have shown that if a/b = c/d, then a + c = b + d directly using algebraic steps.