How can I find the Intersection between 3x -2y =6 and 5x +7y =41?

multiply first equation by 5

15 x -10 y = 30

second equation by -3

-15 x - 21 y = -123
add

0 x - 31 y = -93

y = 93/31 = 3
go back and get x

To find the intersection between two equations, you need to solve the system of equations simultaneously. In this case, you are given the equations:

3x - 2y = 6
5x + 7y = 41

You can solve this system of equations using the method of substitution or elimination. I will explain how to solve it using the elimination method:

Step 1: Multiply both sides of the first equation by 5 and the second equation by 3 to make the coefficients of x equal:

15x - 10y = 30
15x + 21y = 123

Step 2: Subtract the first equation from the second equation to eliminate the x term:

(15x + 21y) - (15x - 10y) = 123 - 30

Simplifying this equation gives:

31y = 93

Divide both sides of the equation by 31:

y = 3

Step 3: Substitute the value of y back into one of the original equations to solve for x. Let's use the first equation:

3x - 2(3) = 6

Simplify this equation:

3x - 6 = 6

Add 6 to both sides:

3x = 12

Divide both sides of the equation by 3:

x = 4

Therefore, the solution to the system of equations is x = 4 and y = 3. This represents the intersection point between the two lines.