I was given this example for solving an equation
ax=b
1/2(ax)=1/2b
(1/2a)x=b/a
x=b/a
I do not understand why you would multiply ax by 1/2. I also do not get why canceling out 1/a times a/1 leads to a transferring to the other side. Thanks for your clarifications!! :)
to solve ax=b for x requires only one step, namely to divide both sides by a
ax/a = b/a
x = b/a
I don't understand the 1/2 nonsense
To solve the equation "ax = b," the goal is to isolate the variable x on one side of the equation.
Now, let's go through each step in the example you provided to understand why we perform certain operations:
1) ax = b
Multiplying both sides of the equation by 1/2:
This step is done to simplify the equation and make it easier to cancel out the coefficient 'a' on the left side. By multiplying both sides of the equation by 1/2, we get:
(1/2) * (ax) = (1/2) * b
Simplifying further:
(1/2)ax = (1/2)b
2) (1/2a)x = b/a
Next, we divide both sides of the equation by 'a' to isolate 'x'. Since 'a' is multiplied by 'x', dividing both sides by 'a' cancels out the 'a' on the left side:
[(1/2a)x]/a = (b/a)/a
Simplifying further:
[(1/2a)x]/a = b/a^2
Now, we have x on the left side, and b/a^2 on the right side.
3) Finally, simplifying further:
[(1/2a)x]/a = b/a^2
Multiplying both sides of the equation by 'a':
[(1/2a)x]/a * a = (b/a^2) * a
This simplifies to:
(1/2a)x = b/a
At this stage, we have x on the left side, so we have successfully isolated the variable.
4) x = b/a
To obtain the final solution, we can multiply both sides of the equation by the reciprocal of (1/2a), which is 2a/1:
x * (2a/1) = (b/a) * (2a/1)
Ultimately, this yields the solution:
2ax = 2b
It's important to note that this final equation, 2ax = 2b, is equivalent to the original equation ax = b. However, the original form ax = b is generally preferred for simplicity unless stated otherwise.
I hope this explanation clears up your confusion! Feel free to ask if you have any further questions.