Simultaneous Equations Part A: Substitution Method

Today I am going to talk about solving simultaneous equations. Simultaneous equations can appear in two forms: linear and linear or linear and quadratic. Luckily for you, the S.A.T only gives linear and linear, however quadratic and linear may also come but the method of solving them on the S.A.T is based on a graph, so it is a lot easier than solving it the proper way. There are two methods of solving simultaneous equations: substitution or elimination. First I shall demonstrate the substitution method.

Consider the linear equations

y = -x + 8…………………eq’n 1
4y = 3x – 24………………eq’n 2

ok, in equation 1 , y = -x + 8, and in eq’n (equation) 2 there exist a term 4y. So substitute y = -x +8 into the term 4y in eq’n 2.

y = -x + 8
4y = 3x – 24

substitute , y = -x + 8 into eq’n 2

4(-x + 8) = 3x – 24…………………now expand this
-4x + 32 = 3x – 24………………….now put like terms on one side
24 + 32 = 3x + 4x…………………...now simplify
56 = 7x……………………………...now solve for x
8 = x………………………………...now sub x into eq’n 1 to get y
y = -(8) + 8
y =0

Therefore the answer for the simultaneous equations are x = 8 and y = 0. But what this really means??? It means that if these two lines were drawn on graph paper, the answer gives the point of intersection. For these two linear equations, the point (8,0) is their point of intersection. This is how the basic simultaneous equations are solved, but the S.A.T doesn’t give a question based only on a quadratic equation. Take a look at this actual question from a S.A.T book.

Line q is given by the equation y = -x + 8, and line r is given by 4y = 3x – 24. If the line r intersects the y-axis at point A, line q intersects the y-axis at point B, and both lines intersect the x-axis at point C, what is the area of the triangle ABC?

Luckily for you it is the last question in the section, and this is as hard as it comes, but to tell you the truth this question is so easy with my technique. Unfortunately this topic covers 3 areas in mathematics, simultaneous equations, area and coordinate geometry. The technique I possess is in the coordinate geometry aspect so I will refer to this exact question when I am doing coordinate geometry.

Solving this question above only gives you point C. when I am doing coordinate geometry I will solve this question. Ok now it is time to look at the other method of solving simultaneous equations: elimination.