simultaneous equations and inequalities (y1)

linear simultaneous equations

simultaneous equations with two unknowns, both to the power of one, with one pair of solutions

can be solved by elimination or substitution

quadratic simultaneous equations

simultaneous equations where one is linear and one is quadratic, with up to two pairs of solutions

solving simultaneous equations by elimination

-multiply one or both of the equations so that the multiple of one of the unknowns matches

-subtract one equation from another to eliminate the equal unknown

-calculate the value of the second unknown using the result

-calculate the value of the first unknown by subtracting the second unknown from one of the equations

-check the results by substituting into the original equations

solving simultaneous equations by substitution

-rearrange one of the equations to make one of the unknowns the subject

-substitute this equation into the other equation

-solve the second equation to find the value of one of the unknowns

-substitute this unknown into the first equation and use that to figure out the value of the other unknown

-check the results by substituting into the oriignal equations

solving quadratic simultaneous equations

-rearrange the linear equation to make one of the unknowns the subject

-substitute this equation into the quadratic equation

-solve the quadratic equation to find the value(s) of one of the unknowns

-substitute the known unknowns into the rearranged linear equation to find the value of the other unknown

-check the results by substituting into the original equations

simultaneous equations on graphs

simultaneous equations can be plotted on graphs- linear equations will be straight lines while quadratic equations will be parabolas

the solution(s) are equal to the point(s) of intersection

linear inequalities

similar to linear equations, but with inequality signs instead
therefore the solutions take ranges of values

quadratic inequalities

similar to quadratic equations, but with inequality signs instead
therefore the solutions take ranges of values

solving linear inequalities

linear inequalities are solved the same way as other linear equations, but the solutions take a range using inequality signs

solving quadratic inequalities

quadratic inequalities are solved the same as other quadratic equations, but the solutions take a range

the range of solutions either take the area between the x-intercepts or outside of them, or above or below the curve

which of these ranges it takes depends on the value of a and the inequality sign, it can be determined by sketching the quadratic equation as a graph

representing solutions to inequalities

inequalities can be:

-drawn on number lines

-written normally ie 5 < x < 7

-written using set notation ie {x: x < 5} ∪ {x: x > 10}

number lines

can be used to represent solutions to inequalities

filled in dots represent <= and >= while empty dots represent < and >

set notation

can be used to represent solutions to inequalities e.x. {x: x < 3}

useful for non-graphically representing values outside of a range rather than in it ie

{x: x < 5} ∪ {x: x > 10} (x is smaller than 5 or bigger than 10, not between them)

inequalities on graphs

inequalities can be plotted on graphs

solid lines represent < and > while dotted lines represent <= and >=

the region of the graph which satisfies the inequalities is shaded

this region is determined using the fact that the solution for each line is on one side of the line for straight lines, or within a range inside or outside of the curve for parabolas