Maths Workshops - Simultaneous Equations and Inequalities

1 Maths WORKSHOPSS imultaneous Equations and InequalitiesBusiness SchoolOutlineRecap of Algebra, Linear and Quadratic FunctionsSimultaneous EquationsInequalitiesApplications in BusinessSummary and ConclusionRevisionSimultaneous EquationsInequalitiesApplicationConclusi onOutlineRecap of Algebra, Linear and Quadratic FunctionsSimultaneous EquationsInequalitiesApplications in BusinessSummary and ConclusionRevisionSimultaneous EquationsInequalitiesApplicationConclusi onVariables, Parameters & Solving EquationsDefinition (Parameters)A parameter is some fixed value, also known as a constant or coefficient.

2 MoreDefinition (Variables)A variable is an unknown value that may change, or vary,depending on the parameter (Solving an equation)We can solve an equation by using mathematical operations torearrange the equation such that the variable is on one side of theequation and the parameters are all on the other side. Example:x=c EquationsInequalitiesApplicationConclusi onLinear functionsDefinition (Linear function)An equation with two variables of the formy=ax+bis called alinear (Independent and dependent variables)The variable on the right hand side of the equation,x, is called theindependent variable and the variable on the left hand side of theequation,y, is called the dependent variable.

3 The dependent variable may also be written asy=f(x)ory=g(x). This notation emphasises thatyis a function ofx, in otherwordsydepends EquationsInequalitiesApplicationConclusi onOutlineRecap of Algebra, Linear and Quadratic FunctionsSimultaneous EquationsInequalitiesApplications in BusinessSummary and ConclusionRevisionSimultaneous EquationsInequalitiesApplicationConclusi onSimultaneous EquationsDefinition ( Simultaneous Equations )If two Equations are both true at the same time, they are calledsimultaneous system of two Simultaneous Equations :y= 4x2x+y= 6 Definition (Solution)To solve a system of Simultaneous Equations we need to find valuesof the variables that satisfy all Equations in the EquationsInequalitiesApplicationConclusi onSimultaneous EquationsDefinition (Solution)To solve a system of Simultaneous Equations we need to find valuesof the variables that satisfy all Equations in the this is the point where the two lines cross:xy246-11234y= 4x2x+y= 6 RevisionSimultaneous EquationsInequalitiesApplicationConclusi onHow to solve systems of Equations ?

4 The general approach consists of 3 steps:1. Manipulate the Equations to find an expression in terms ofone variable Solve the equation for that one variable3. Use that solution in one of the original Equations to find theother are two main ways to manipluate the Equations in step 1:Definition (Substitution Method)Substitute one equation into (Elimination Method)Add or subtract a multiple of one equation from the EquationsInequalitiesApplicationConclusi onSubstitution MethodWe can use the 3 step approach to solve the following system:y= 4x(1)2x+y= 6(2)1.

5 Substitute Equation (1) into Equation (2):2x+ 4x= 6(substitutingy= 4x)6x= 62. Solve this equation forx:6x 16= 6 16(divide both sides by6)x= 13. Use this solution,x= 1, in Equation (1) to findy:y= 4x= 4 1 = 4 RevisionSimultaneous EquationsInequalitiesApplicationConclusi onElimination Method3 step approach using the elimination method:y= 4x(1)2x+y= 6(2)1. Eliminateyin Equation (2) by subtracting (1) from (2):2x+y y= 6 4x2x= 6 4x(no longer anyy s)6x= 62. Using exactly the same approach as in the substitutionmethod we solve to findx= As before, we substitutex= 1back into Equation (1) to findy= EquationsInequalitiesApplicationConclusi onSimultaneous Equations Your Turn.

6 Solve the following system of equations2x+y= 8(3)x+y= 6(4) elimination method and subtract (4) from (3):2x+y (x+y) = 8 6x= 2OR the substitution method by rearranging (4) to gety= 6 xand substituting this into (3):2x+ (6 x) = 8x= need to solve forxas we can see directly thatx= 2in (4) to findy= 6 2 = EquationsInequalitiesApplicationConclusi onSimultaneous Equations Your Turn..Solve the following system of equations2x+y= 8(3)x+y= 6(4)1. The elimination method and subtract (4) from (3):2x+y (x+y) = 8 6x= 2OR the substitution method by rearranging (4) to gety= 6 xand substituting this into (3):2x+ (6 x) = 8x= need to solve forxas we can see directly thatx= 2in (4) to findy= 6 2 = EquationsInequalitiesApplicationConclusi onSimultaneous Equations Your Turn.

7 Solve the following system of equations2x+y= 8(3)x+y= 6(4)1. The elimination method and subtract (4) from (3):2x+y (x+y) = 8 6x= 2OR the substitution method by rearranging (4) to gety= 6 xand substituting this into (3):2x+ (6 x) = 8x= need to solve forxas we can see directly thatx= 2in (4) to findy= 6 2 = EquationsInequalitiesApplicationConclusi onSimultaneous Equations Your Turn..Solve the following system of equations2x+y= 8(3)x+y= 6(4)1. The elimination method and subtract (4) from (3):2x+y (x+y) = 8 6x= 2OR the substitution method by rearranging (4) to gety= 6 xand substituting this into (3):2x+ (6 x) = 8x= 22.

8 No need to solve forxas we can see directly thatx= Usex= 2in (4) to findy= 6 2 = EquationsInequalitiesApplicationConclusi onGraphical Example In the Algebra Workshop we showed how to graph linearfunctions. The solution of a system of Equations can be graphicallyrepresented as the point of intersection of the two (Intersection)The intersection is the point at which two lines (Cartesian coordinates)It is sometimes useful to use the cartesian coordinate system torefer to points in the 2-dimensional plane. Instead of writingx= 2andy= 4we instead write as(x, y) = (2,4)or just refer to thepoint(2,4).

9 MoreRevisionSimultaneous EquationsInequalitiesApplicationConclusi onGraphing Simultaneous equationsy= 4x2x+y= 6 y= 4xis simple to plot, it goes through the origin(x, y) = (0,0)and has slope equal to 4. 2x+y= 6is a bit tricky. Whenx= 0the intercept isy= 6and wheny= 0 = 2x= 6orx= 3, so the line passesthrough the two points (0,6) and (3,0):xy246-11234y= 4xy= 2x+ 6 Solution:(x, y) = (1,4)RevisionSimultaneous EquationsInequalitiesApplicationConclusi onGraphing Simultaneous equationsy= 4x2x+y= 6 y= 4xis simple to plot, it goes through the origin(x, y) = (0,0)and has slope equal to 4.

10 2x+y= 6is a bit tricky. Whenx= 0the intercept isy= 6and wheny= 0 = 2x= 6orx= 3, so the line passesthrough the two points (0,6) and (3,0):xy246-11234y= 4xy= 2x+ 6 Solution:(x, y) = (1,4)RevisionSimultaneous EquationsInequalitiesApplicationConclusi onGraphing Simultaneous equationsy= 4x2x+y= 6 y= 4xis simple to plot, it goes through the origin(x, y) = (0,0)and has slope equal to 4. 2x+y= 6is a bit tricky. Whenx= 0the intercept isy= 6and wheny= 0 = 2x= 6orx= 3, so the line passesthrough the two points (0,6) and (3,0):xy246-11234y= 4xy= 2x+ 6 Solution:(x, y) = (1,4)RevisionSimultaneous EquationsInequalitiesApplicationConclusi onGraphing Simultaneous equationsy= 4x2x+y= 6 y= 4xis simple to plot, it goes through the origin(x, y) = (0,0)and has slope equal to 4.