Find three solutions of each equation.
4x + 3y = 14
and
6x - 5y = 23
pick any x, say 5, and plug it in:
4*5 + 3y = 14
now solve for y
20 + 3y = 14
3y = -6
y = -2
So, (5,-2) is a solution
Now just do some more. You can also pick a y value and then solve for x.
To find three solutions for each equation, we can start by assigning arbitrary values to one of the variables and then solving for the other variable.
For the equation 4x + 3y = 14:
1. Let's assign a value to x. Let's say x = 0.
Substitute this value into the equation:
4(0) + 3y = 14
0 + 3y = 14
3y = 14
Divide both sides by 3:
y = 14/3
Therefore, one solution is (x, y) = (0, 14/3).
2. Let's assign another value to x. Let's say x = 2.
Substitute this value into the equation:
4(2) + 3y = 14
8 + 3y = 14
3y = 14 - 8
3y = 6
Divide both sides by 3:
y = 6/3
y = 2
Therefore, another solution is (x, y) = (2, 2).
3. Let's assign one more value to x. Let's say x = -2.
Substitute this value into the equation:
4(-2) + 3y = 14
-8 + 3y = 14
3y = 14 + 8
3y = 22
Divide both sides by 3:
y = 22/3
Therefore, a third solution is (x, y) = (-2, 22/3).
For the equation 6x - 5y = 23, you can follow the same process to find three solutions.
To find the solutions to each equation, we can use the method of substitution or elimination.
Let's start with the first equation:
1. Substitution method:
We solve one equation for one variable and substitute it into the other equation.
Step 1: Solve for x in terms of y.
4x + 3y = 14
4x = 14 - 3y
x = (14 - 3y) / 4
Step 2: Substitute x in terms of y into the second equation.
6x - 5y = 23
6((14 - 3y) / 4) - 5y = 23
Multiply through by 4 to avoid fractions.
6(14 - 3y) - 20y = 92
84 - 18y - 20y = 92
-38y = 8
y = -8/38 = -4/19
Step 3: Substitute y back into the equation for x.
x = (14 - 3(-4/19)) / 4
x = (14 + 12/19) / 4
x = (266/19) / 4
x = 266/19 * 1/4
x = 133/76
The first solution is x = 133/76, y = -4/19.
Now, let's find two more solutions:
2. Let y = 0:
4x + 3(0) = 14
4x = 14
x = 14/4
x = 7/2
The second solution is x = 7/2, y = 0.
3. Let x = 0:
4(0) + 3y = 14
3y = 14
y = 14/3
The third solution is x = 0, y = 14/3.
Now, let's move on to the second equation:
1. Substitution method:
We solve one equation for one variable and substitute it into the other equation.
Step 1: Solve for x in terms of y.
6x - 5y = 23
6x = 23 + 5y
x = (23 + 5y) / 6
Step 2: Substitute x in terms of y into the second equation.
4((23 + 5y) / 6) + 3y = 14
Multiply through by 6 to avoid fractions.
4(23 + 5y) + 18y = 84
92 + 20y + 18y = 84
38y = -8
y = -8/38 = -4/19
Step 3: Substitute y back into the equation for x.
x = (23 + 5(-4/19)) / 6
x = (23 - 20/19) / 6
x = (437/19) / 6
x = 437/19 * 1/6
x = 437/114
The first solution is x = 437/114, y = -4/19.
Now, let's find two more solutions:
2. Let y = 0:
6x - 5(0) = 23
6x = 23
x = 23/6
The second solution is x = 23/6, y = 0.
3. Let x = 0:
6(0) - 5y = 23
-5y = 23
y = -23/5
The third solution is x = 0, y = -23/5.
Therefore, the three solutions for the first equation are (133/76, -4/19), (7/2, 0), and (0, 14/3).
And the three solutions for the second equation are (437/114, -4/19), (23/6, 0), and (0, -23/5).