Use simultaneous equations to find the points where the following lines cross: 3x - y = 4 and x^2 + 7y = 5
The points where the lines cross are the points where the two equations equal each other. To do this we solve simultaneous equations. Call equations as follows:(a) 3x - y = 4(b) x2 + 7y = 5 First take (a) and rearrange to get y in terms of x + y to both sides and -4 from both sides. This gives y = 3x - 4 and call this equation (a.2). Input equation (a.2) into (b) to replace all the y's Gives x2+ 7(3x - 4) = 5, times out the bracket. This gives x2+ 21x - 28 = 5, -5 from both sides. This gives x2+ 21x - 33 = 0 and call this equation (b.2). Then use the quadratic equation (-b +- sqrt(b2- 4ac))/2a) to solve (b.2). In (b.2), a = 1, b = 21, c = -33 for the quadratic equation. This gives x = (-21 +- sqrt(212- 41-33))/21) = -10.5 +- 11.968 = -10.5 +- 12. This gives two possible solutions to x, x1= 1.5 and x2= - 22.5. These are the x coordinates for the crossing points. Plug in both solutions for x into (a) to give the two correlating y coordinates. For x1 in (a), gives y1 = 3(1.5) -4 = 0.5. For x2 in (a), gives y2 = 3*(-22.5) -4 = -71.5. This means the two coordinates where the lines cross are (1.5, 0.5) and (-22.5,-71.5). We can check that these are the correct answers by plugging them back into equations (a) and (b). For (a), 3x1- y1 = 3(1.5) - 0.5 = 4 which is correct For (b), 3x2- y2= 3(-22.5) - (-71.5) = 4 which is also correct!
