Question 3.2: Given the simultaneous equations 2x + 3y = 12.5 (1) −x + 2y ......

(a) In this example, neither the x- nor the y-terms are the same. However, all terms on both sides of any equation may be multiplied by a constant without affecting the solution of the equation. So, if equation (2) is multiplied by 2, the x-terms in both equations will be the same with opposite signs. Then, proceed as in Worked Example 3.1

Step 1: Eliminate x from the system of equations by multiplying equation (2) by 2 and then add the equations; 2x and –2x cancel to leave a single equation in one unknown, y:

2x + 3y = 12.5           (1) as given
−2x + 4y = 12            (2) × 2
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0 + 7y = 24.5              adding the two equations
y =\frac{24.5}{7} = 3.5             solving for y

Step 2: Solve for the value of x by substituting y=3.5 into either equation (1) or equation (2):

−x + 2(3.5) = 6       substituting y = 3.5 into (2) to find the value of x
−x = 6 − 7
−x = −1 → x = 1

Step 3: Check the solution x = 1, y = 3.5, by substituting these values into equations (1) and (2) and confirm that both equations balance.

• Substitute x = 1 and y = 3.5 into equation (1):

2x + 3y = 12.5
2(1) + 3(3.5) = 12.5       substituting in x = 1 and y = 3.5
2 + 10.5 = 12.5
12.5 = 12.5                      so (1) balances and x = 1, y = 3.5 is a solution

• Substitute x = 1 and y = 3.5 into equation (2):

−x + 2y = 6
−(1) + 2(3.5) = 6     substituting in x = 1 and y = 3.5
−1 + 7 = 6
6 = 6                         so (2) balances and x = 1, y = 3.5 is a solution
Therefore, the solution of equations (1) and (2) is at the point of intersection of the lines represented by equations (1) and (2), as shown in Figure 3.2.

(b) The two lines are plotted in Figure 3.2. The point of intersection is the solution. The coordinates of this point are x = 1 and y = 3.5. In this case it is a unique solution: that is, the lines intersect at only one point. This point is on the first line, so it satisfies equation (1), and also on the second line, so it satisfies equation (2).