Linear Equations

To solve the system of equations, we can use the substitution method. First, solve one of the equations for one of the variables. For example, we can solve the second equation for (y): (y = 4 - 2x). Then, substitute this expression for (y) into the other equation: (3x - 2(4 - 2x) = 1). This gives us (3x - 8 + 4x = 1), which simplifies to (7x = 9). Dividing both sides by 7, we get (x = 9/7). Substituting this value of (x) back into the second equation, we get (2(9/7) + y = 4), which simplifies to (y = 1). Therefore, the solution to the system of equations is ((x, y) = (2, 1)).

To solve the system of equations, we can use the elimination method. First, multiply the second equation by 3 to get (6x - 3y = 3). Then, add this equation to the first equation to get (10x = 14). Dividing both sides by 10, we get (x = 14/10 = 7/5). Substituting this value of (x) back into the first equation, we get (4(7/5) + 3y = 11), which simplifies to (y = 3). Therefore, the solution to the system of equations is ((x, y) = (2, 3)).

To solve the system of equations, we can use the addition method. First, add the two equations together to get (3x = 9). Dividing both sides by 3, we get (x = 3). Substituting this value of (x) back into the first equation, we get (3 - y = 2), which simplifies to (y = 1). Therefore, the solution to the system of equations is ((x, y) = (3, 1)).

To solve the system of equations, we can use the elimination method. First, multiply the second equation by 2 to get (6x - 4y = 2). Then, add this equation to the first equation to get (8x - y = 10). Dividing both sides by 8, we get (x - (1/8)y = (5/4)). Now, multiply the second equation by 3 to get (9x - 6y = 3). Then, add this equation to the first equation to get (17x - 7y = 8). Dividing both sides by 17, we get (x - (7/17)y = (8/17)). Now, we have two equations in two variables: (x - (1/8)y = (5/4)) and (x - (7/17)y = (8/17)). We can solve this system of equations using the substitution method. First, solve one of the equations for (x). For example, we can solve the first equation for (x): (x = (5/4) + (1/8)y). Then, substitute this expression for (x) into the other equation: ((5/4) + (1/8)y - (7/17)y = (8/17)). This gives us (17(5/4) + 17(1/8)y - 17(7/17)y = 17(8/17)), which simplifies to (y = 3). Substituting this value of (y) back into the first equation, we get (x - (1/8)(3) = (5/4)), which simplifies to (x = 2). Therefore, the solution to the system of equations is ((x, y) = (2, 3)).

To solve the system of equations, we can use the substitution method. First, solve one of the equations for one of the variables. For example, we can solve the first equation for (x): (x = 5 - 2y). Then, substitute this expression for (x) into the other equation: (2(5 - 2y) - y = 3). This gives us (10 - 4y - y = 3), which simplifies to (-5y = -7). Dividing both sides by (-5), we get (y = 7/5). Substituting this value of (y) back into the first equation, we get (x + 2(7/5) = 5), which simplifies to (x = 1). Therefore, the solution to the system of equations is ((x, y) = (1, 2)).

To solve the system of equations, we can use the elimination method. First, multiply the second equation by 2 to get (4x + 2y = 8). Then, add this equation to the first equation to get (7x = 15). Dividing both sides by 7, we get (x = 15/7). Substituting this value of (x) back into the second equation, we get (2(15/7) + y = 4), which simplifies to (y = 4). Therefore, the solution to the system of equations is ((x, y) = (3, 4)).

To solve the system of equations, we can use the addition method. First, add the two equations together to get (2x = 6). Dividing both sides by 2, we get (x = 3). Substituting this value of (x) back into the first equation, we get (3 + y = 5), which simplifies to (y = 2). Therefore, the solution to the system of equations is ((x, y) = (3, 2)).

To solve the system of equations, we can use the substitution method. First, solve one of the equations for one of the variables. For example, we can solve the first equation for (x): (x = (7 - 3y)/2). Then, substitute this expression for (x) into the other equation: (4((7 - 3y)/2) - y = 5). This gives us (14 - 6y - y = 5), which simplifies to (-7y = -9). Dividing both sides by (-7), we get (y = 9/7). Substituting this value of (y) back into the first equation, we get (2x + 3(9/7) = 7), which simplifies to (x = 2). Therefore, the solution to the system of equations is ((x, y) = (2, 1)).

Katyayana's method for solving simultaneous equations is the method of elimination.

To solve the equation 3x + 5 = 17, we need to isolate the variable x on one side of the equation. We can do this by subtracting 5 from both sides of the equation, which gives us 3x = 12. Then, we can divide both sides of the equation by 3, which gives us x = 4.

To solve the equation 3x + 5 = 0, we need to isolate the variable x on one side of the equation. We can do this by subtracting 5 from both sides of the equation, which gives us 3x = -5. Then, we can divide both sides of the equation by 3, which gives us x = -5/3.

To solve the equation 2x - 7 = 1, we need to isolate the variable x on one side of the equation. We can do this by adding 7 to both sides of the equation, which gives us 2x = 8. Then, we can divide both sides of the equation by 2, which gives us x = 4.

To solve the equation 4x + 2 = 10, we need to isolate the variable x on one side of the equation. We can do this by subtracting 2 from both sides of the equation, which gives us 4x = 8. Then, we can divide both sides of the equation by 4, which gives us x = 2.

To solve the equation 5x - 3 = 7, we need to isolate the variable x on one side of the equation. We can do this by adding 3 to both sides of the equation, which gives us 5x = 10. Then, we can divide both sides of the equation by 5, which gives us x = 2.

To solve the equation 6x + 1 = 13, we need to isolate the variable x on one side of the equation. We can do this by subtracting 1 from both sides of the equation, which gives us 6x = 12. Then, we can divide both sides of the equation by 6, which gives us x = 2.