Helpsheet Giblin Eunson Library LINEAR EQUATIONS Use this sheet to help you: Solve linear equations containing one unknown Recognize a linear function, and identify its slope and intercept parameters Recognize and solve linear simultaneous equations with two unknowns by both algebraic and graphical methods Author: Carter, D., Design and Layout: Pesina, J.
Operations on an equation Any operation can be performed on an equation, as long as the operation is performed on the whole of both sides of the equation. e.g. 1. x + 7 = 12 subtract 7 from both sides, which gives x + 7 7 = 12 7, which simplifies to x = 5 2. multiply both sides by 5, which gives which simplifies to x = 30 3. subtract 7 from both sides, which gives,multiply both sides by x x + 5 = 3x, subtract x from both sides x x +5= 3x x 5 = 2x, divide both sides by 2 Page 1
Variables and Parameters The known values in an equation are called constants, coefficients or parameters. Sometimes when discussing an equation, we may want to look at it in a general way, without specifying values for the parameters, so we assign letters rather than numbers. By convention we usually assign letters from the beginning of the alphabet to distinguish them from the variables (unknowns), which are usually assigned letters from the end of the alphabet. Sometimes subscripts or superscripts are used as part of the labelling of variables or parameters For example q D and q S, where the superscripts D and S distinguish quantity demanded and quantity supplied. Similarly we might write t0 and t1 where the subscripts 0 and 1 distinguish an initial tax rate from the new tax rate. The linear equation and its solution Any linear equation with one unknown has the general form ax + b = c where x is the variable and a, b and c are the unspecified parameters (with a 0). Its solution is Linear functions y = ax + c is an example of a linear equation with two variables, x and y, and two parameters, a and c. A linear equation which contains two variables is called a linear function. A way of denoting a function is y = f(x) or y = g(x) or y = φ(x). The variable which appears on the right hand side of the equation (x) is called the independent variable and the variable on its own on the left hand side of the equation (y) is called the dependent variable. Graphs of linear functions The independent variable, x, is measured along the horizontal axis, and the dependent variable, y, is measured along the vertical axis. The point where the two axes meet is called the origin. Every point on the x-y plane corresponds to a unique pair of values of x and y. These values are known as the coordinates of the point. Page 2
The slope and y-intercept of a linear function The slope or gradient of a graph can be measured looking at the increase in y that results when x increases by one unit. When given a linear function in the form y = ax + c, the slope or gradient is a. If a is positive the graph slopes upwards from left to right and if a is negative, the graph slopes downward from left to right. If a = 0, then ax = 0, which leaves y = b. The resulting graph is a horizontal line, with an intercept of b on the y-axis. The equation x = a, results in a vertical line graph, with an x-intercept of a on the x axis. The y-intercept (the point where the line cuts the y axis) is found at the point where x = 0, therefore when given a linear function in the form y = ax + c, the y-intercept is c. (To find the x-intercept, y = 0). A linear function may also be given in the form ax + by + c = 0. Simultaneous linear equations If two equations, such as y = 4x and 2x + y = 6, are both true at the same time, they are called simultaneous equations. What is needed here is to find the pair of values (x, y) that satisfies both of these equations. This is what is meant by solving simultaneous equations. In the example given we can substitute the value for y (4x) from the first equation into the second equation to get 2x + 4x = 6 6x = 6 x = 1. Once you have found a value for x, you can substitute that value into one of the equations to find the value for y. Substituting into the first equation gives us y = 4 x 1 y = 4. So the pair of values is (1, 4). There are other techniques for solving simultaneous equations, which you may have learned previously. You may prefer your method to the one shown. Graphical solution of simultaneous equations The solution to a pair of simultaneous linear equations is given by the coordinates of the point of intersection of their graphs. If the graphs are parallel i.e. they have the same gradient (slope), they will never intersect and therefore will not have a solution. If you are given three or more equations with two unknowns, unless there is unique point of intersection for the equations there will be no pair of values that satisfy the equations simultaneously. Page 3
Three linear equations with three unknowns Solving three linear equations with three unknowns is an extension of the method used to solve two simultaneous equations. e.g. z = x + y + 4 (1) z = 2x y (2) z = 3x 4y (3) Step 1: let (1) = (2) x + y + 4 = 2x y x = 2y + 4 (4) Step 2: let (2) = (3) 2x y = 3x 4y x = 3y (5) Step 3: let (4) = (5) 2y + 4 = 3y y = 4 Step 4: Substitute y = 4 into (5) x = 3(4) x = 12 Step 5: Substitute y = 4 and x = 12 into (1) (* could substitute into (1), (2) or (3)) z = 12 + 4 + 4 z = 20 So our solution is x = 12, y = 4 and z = 20. You can check in the other two equations to see that this is the correct solution. Page 4