How can you tell if these three equations have a common solution?
3x - 5y = 1
4x - y = 24
2x - 3y = 2
Please help ?
intersect (solve) any two of them, then sub in that solution into the third.
If it satisfies the third equation, then the solution is "common"
is the common solution then (7,4) ?
To determine if a system of equations has a common solution, we need to check if the equations are consistent and dependent. If the equations are consistent (meaning they have at least one solution) and dependent (meaning there are infinitely many solutions), then they have a common solution.
One way to solve this is by using a method called elimination, where we eliminate one variable at a time and solve for the remaining variables.
Let's begin by eliminating the variable y:
1. Start with the first equation: 3x - 5y = 1
2. Multiply the second equation by 5 to make the coefficients of y in both equations the same: 20x - 5y = 120
3. Subtract the second equation from the first equation: (3x - 5y) - (20x - 5y) = 1 - 120
4. Simplify: -17x = -119
5. Divide both sides by -17 to solve for x: x = 7
Now that we have the value of x, we can substitute it back into any of the original equations to solve for y. Let's use the third equation:
6. Substitute x = 7 into the third equation: 2(7) - 3y = 2
7. Simplify: 14 - 3y = 2
8. Subtract 14 from both sides: -3y = -12
9. Divide both sides by -3 to solve for y: y = 4
Therefore, the solution to the system of equations is x = 7 and y = 4. Since we were able to find a unique solution for both variables, the equations have a common solution.