TI Nspire CX Tip: How to Find Points of Intersection

Tip 2: How to find the intersection coordinates of lines

You will need
  • - two straight lines in the drawing;
  • - equations of two straight lines.
Instruction
If the lines are already drawn on the graph, find the solution graphically. To do this, continue both or one of the lines so that they intersect. Then mark the point of intersection and lower the perpendicular from it to the abscissa axis (as a rule, oh).
Using the scale of tick marks marked on the axis, find the value of x for this point. If it is on the positive direction of the axis (to the right of the zero mark), then its value will be positive, otherwise - negative.
Similarly, find the ordinate of the intersection point. If the projection of a point is above the zero mark, it is positive; if it is below, it is negative. Write the coordinates of the point in the form (x, y) - this is the solution to the problem.
If the lines are given in the form of the formulas y = kx + b, you can also solve the problem graphically: draw lines on the grid and find the solution using the method described above.
Try to find a solution to the problem using these formulas. To do this, make a system of these equations and solve it. If the equations are given in the form y = kx + b, simply equate both parts with x and find x. Then substitute the value of x in one of the equations and find y.
You can find a solution way Kramer. In this case, bring the equations to the form A1x + V1y + C1 = 0 and A2x + V2y + C2 = 0. According to Kramer’s formula, x = - (C1B2-C2B1) / (A1B2-A2B1), and y = - (A1C2-A2C1) / (A1B2-A2B1). Note that if the denominator is zero, then the straight lines are parallel or coincide and, accordingly, do not intersect.
If you are given straight lines in space in a canonical form, before you begin your search for a solution, check to see if the straight lines are parallel. To do this, estimate the coefficients before t if they are proportional, for example, x = -1 + 3t, ​​y = 7 + 2t, z = 2 + t and x = -1 + 6t, y = -1 + 4t, z = -5 + 2t, then the lines are parallel. In addition, straight lines can interbreed, in which case the system will have no solution.
If you find out that the lines intersect, find the point of their intersection. First equate the variables from different straight lines, conventionally replacing t with u for the first straight line and v for the second straight line. For example, if you are given straight lines x = t-1, y = 2t + 1, z = t + 2 and x = t + 1, y = t + 1, z = 2t + 8, you will get expressions like u-1 = v +1, 2u + 1 = v + 1, u + 2 = 2v + 8.
Express u from one equation, substitute it into another, and find v (in this problem, u = -2, v = -4).

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Images: Tip 2: How to find the intersection coordinates of lines Tip 2: How to find the intersection coordinates of lines
Images: Tip 2: How to find the intersection coordinates of lines Tip 2: How to find the intersection coordinates of lines
Images: Tip 2: How to find the intersection coordinates of lines Tip 2: How to find the intersection coordinates of lines