solve each system using elimination.
1.) 6x - 3y =15
7x + 14y =10
Tell whether the system has one solution, infinitely many solutions, or no solution.
1.) 9x +8y =15
9x +8y =30
2.) 5x - 3y =10
10x + 6y = 20
2) the last is wrong.
2. one solution
6x -3y = 15
7x + 14 y = 10
14(6x - 3y = 15)
3(7x + 14y = 10)
84x - 42y = 210
21x + 42 y = 30
105x = 240
x = 16/7
y = -3/7
(16/7, - 3/7)
1. No Solution
2. Infinitely many solutions
To solve a system of equations using elimination, the goal is to eliminate one variable by adding or subtracting the equations. Here's how to solve each system using elimination:
1.) 6x - 3y = 15
7x + 14y = 10
To eliminate the y-variable, let's multiply the first equation by 2. This will make the y coefficients in both equations cancel each other out when added together.
Multiply the first equation by 2:
12x - 6y = 30
Now, write down the modified system:
12x - 6y = 30
7x + 14y = 10
Add the equations together to eliminate the y-variable:
(12x - 6y) + (7x + 14y) = 30 + 10
19x + 8y = 40
Simplify the equation:
19x + 8y = 40
So, the system of equations simplifies to:
19x + 8y = 40 (Equation 1)
7x + 14y = 10 (Equation 2)
Now, let's determine the number of solutions for this system:
Since the system still has two variables (x and y) and not enough information to determine their values, we cannot yet say whether there is one solution, infinitely many solutions, or no solution. We need one more equation or condition to uniquely determine the values of x and y.
Now let's solve the second system:
2.) 5x - 3y = 10
10x + 6y = 20
To eliminate the y-variable, let's multiply the first equation by 2 and the second equation by 3:
Multiply the first equation by 2:
10x - 6y = 20
Multiply the second equation by 3:
30x + 18y = 60
Now, write down the modified system:
10x - 6y = 20
30x + 18y = 60
Add the equations together to eliminate the y-variable:
(10x - 6y) + (30x + 18y) = 20 + 60
40x + 12y = 80
Divide the equation by 4 to simplify:
10x + 3y = 20
So, the system of equations simplifies to:
10x + 3y = 20 (Equation 1)
10x + 6y = 20 (Equation 2)
Now, let's determine the number of solutions for this system:
Notice that both equations are identical when multiplied by 3. This means that the two equations represent the same line in the coordinate plane. Therefore, the system has infinitely many solutions.
To summarize:
1.) The system 6x - 3y = 15 and 7x + 14y = 10 simplifies to 19x + 8y = 40 and 7x + 14y = 10. However, we need one more equation or condition to determine if there is one solution, infinitely many solutions, or no solution.
2.) The system 5x - 3y = 10 and 10x + 6y = 20 simplifies to 10x + 3y = 20 and 10x + 6y = 20. Since the two equations represent the same line, the system has infinitely many solutions.