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## Simultaneous Linear Equations

Consider two linear equations in two variables, x and y, such as

2x - 3y = 4

3x + y = 1

Instead of one equation in one unknown, we have here two equations and two unknowns. In order to find a solution for this pair of equations, the unknown numbers x and y have to satisfy both equations. Hence, we call this system or pair of equations or simultaneous equations. We now focus on various methods of solving simultaneous equations.

## Intersection Point of a Line with a Horizontal or Vertical Line

We first consider the special cases of solving a pair of simultaneous linear equations when one of the two lines is either horizontal (ay = b) or vertical (cx = d); the solution in these cases is easily found by substitution.

### Example 1A.

Case one: One line is horizontal (ay = b):

 Equation 1: x + y = Equation 2: y =

Click on the step buttons to show each solving stage.

First, solve Equation 2 for y:    y =

Next, substitute y = into Equation 1 and solve for x:

x + ( ) =

x = (2 d.p.)

### Exercise 1A.

Now try solving a few on your own:

 Solve these simultaneous linear equations, rounding any answers to two decimal places: x + y = y = y = x = (2 d.p.)

### Example 1B.

Case two: One line is vertical (cx = d). The method is exactly the same as for a vertical line.

 Equation 1: x + y = Equation 2: x =

Click on the step buttons to show each solving stage.

First, solve Equation 2 for x:    x =

Next, substitute x = into Equation 1 and solve for y:

( ) + y =

y = (2 d.p.)

### Exercise 1B.

Practice this method with a few exercises:

 Solve these simultaneous linear equations, rounding any answers to two decimal places: x + y = x = x = y = (2 d.p.)

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