Find three ordered pairs (a,b) that satisfy the equation 3a - 5b = 9.
So far, I got (3,0). I'm having trouble finding others. Does anyone have suggestions on other easy ways to find ordered pairs besides guess-and-check? Thank you!
change it to
3a = 5b + 9
a = (5/3)b + 3
pick any multiple for b , eg. b = -3,3,6,-6 etc and out comes the a as an integer.
Your point is when b = 0 (also a multiple of 3 btw)
To find more ordered pairs that satisfy the equation 3a - 5b = 9, you can use a method called substitution. Here's how to do it:
Step 1: Solve the equation for one variable in terms of the other. In this case, let's solve for 'a'.
3a - 5b = 9
3a = 5b + 9
a = (5b + 9)/3
Step 2: Choose any value for 'b' and substitute it into the equation to find the corresponding value for 'a'. Let's choose 'b = 1'.
a = (5(1) + 9)/3
a = (5 + 9)/3
a = 14/3
So, one ordered pair that satisfies the equation is (14/3, 1).
Step 3: Repeat the process to find more ordered pairs.
Let's try 'b = 2':
a = (5(2) + 9)/3
a = (10 + 9)/3
a = 19/3
Another ordered pair that satisfies the equation is (19/3, 2).
Let's try 'b = -1':
a = (5(-1) +9)/3
a = (-5 + 9)/3
a = 4/3
One more ordered pair that satisfies the equation is (4/3, -1).
Therefore, the three ordered pairs that satisfy the equation 3a - 5b = 9 are (14/3, 1), (19/3, 2), and (4/3, -1).
To find ordered pairs that satisfy the equation 3a - 5b = 9 without guess-and-check, you can use a systematic approach. One common method is to set one variable equal to a constant value and solve for the other variable.
Step 1: Choose a value for either variable. Let's choose a value for b.
Let's set b = 0, as you have already done.
Step 2: Substitute the chosen value into the equation and solve for the other variable.
Substituting b = 0 into the equation 3a - 5b = 9, we get:
3a - 5(0) = 9
3a - 0 = 9
3a = 9
a = 3
So we found one ordered pair (a,b) = (3,0).
Step 3: Repeat the steps for different values of the chosen variable.
Now, let's choose a value for a. Let's set a = 0.
Substituting a = 0 into the equation 3a - 5b = 9, we get:
3(0) - 5b = 9
0 - 5b = 9
-5b = 9
b = -9/5
So another ordered pair is (a,b) = (0, -9/5).
Step 4: Repeat the steps one more time.
Let's choose another value for a. Let's set a = 1.
Substituting a = 1 into the equation 3a - 5b = 9, we get:
3(1) - 5b = 9
3 - 5b = 9
-5b = 6
b = -6/5
So another ordered pair is (a,b) = (1, -6/5).
In summary, the three ordered pairs that satisfy the equation 3a - 5b = 9 are: (3,0), (0, -9/5), and (1, -6/5).