Systems of Equations and Inequalities
Chapter 7Systems of Equations and Inequalities7-1 Solving
Systems by GraphingCombining two or more equations together
(usually joined by set brackets) forms a system of equationsThe
solution to a system of equations is the point where they all
intersectThe solution is the point that you can plug into all of
the equations and it makes all of them trueEx1. Is (4, -3) a
solution of
One way to solve a system of equations is to graph all of the
equations and determine the intersection pointThis way is only
effective if the answers are integers
One question per graphSolve each system by graphingEx2.
Ex3.
If the linear equations are parallel, there will be no
solutionWrite no solution as your answerIf the linear equations are
the same line, there will be infinitely many solutionsWrite
infinitely many solutions as you answerSee the bottom of page 342
for graphic depictions Graphing Calculators and Systems of
EquationsIf you input both equations (in slope-intercept form) into
a graphing calculator, it will give you the solutionPress Y= Input
one equation in Y1 and the other in Y2Press GRAPHPress 2nd and then
TRACE (now CALC)Choose intersect (choice 5)Your blinky light will
be on one of the lines (called first curve), so hit ENTERThe blinky
light will jump to the other line (called second curve), so hit
ENTERNow you need to guess where the intersection is, so use the
left or right arrow keys to put the blinky light near the
intersection and press ENTERThe solution is at the bottom of your
screenUse the graphing calculator to find the solution to each of
the following systemsEx1.
Ex2.
7-2 Solving Systems Using SubstitutionAnother method for solving
systems of equations is substitutionThis method is most useful when
the solution is not an integer and at least one of the equations is
in slope-intercept formWith substitution you are substituting an
equivalent expression for one of the variablesSolve one of the
equations for one of the variablesIt does not matter which one you
choose, you will get the same answer, so choose the simplest
oneThis step may already be done for you Plug the expression you
found into the OTHER equation for that variableNow you have a
multi-step equation to solveYou MUST show all of the appropriate
workSolve each system of equations using substitutionEx1. Ex2.
Ex3.
7-3 Solving Systems Using EliminationElimination is another way
to solve systems of equationsIt is best used when one or both of
the equations are in standard form and/or the answer is not
integers1) Write the equations in standard form (align the
variables)2) Multiply through one or both of the equations so that
the coefficients for one of the variables (it doesnt matter which
one) are oppositesThis step may be done for you at times3) Add the
like terms (straight down the columns), eliminating one of the
variables4) Solve for both variablesSolve each system using
eliminationEx1.
Ex2.
Ex3.
7-4 Applications of Linear SystemsSystems frequently occur in
real-world situationsYou will have to define your variables, write
your system of equations, determine which method is best for
solving, and then solve (showing all of the appropriate work)Ex1. A
chemist has one solution that is 50% acid. She has another solution
that is 25% acid. How many liters of each type of acid solution
should she combine to get 10 liters of 40% acid solution?Ex2.
Suppose you have a typing service. You buy a personal computer for
$1750 on which to do your typing. You charge $5.50 per page for
typing. Expenses are $0.50 per page for ink, paper, electricity,
etc. How many pages must you type to break even?7-5 Linear
InequalitiesJust like graphing inequalities on a number line,
graphing inequalities on a coordinate plane will require shadingThe
shading indicates all of the possible solutionsOn a number line
< and > mean to use an open circleOn a coordinate plane <
and > mean to use dashed linesOn a number line < and >
mean to use a closed circleOn a coordinate plane < and > mean
to use solid linesIf the inequality is in slope-intercept form,
then < and < mean to shade below the lineIf the inequality is
in slope-intercept form, then > and > mean to shade above the
lineBe sure you can still tell if the line is solid or dashed!If
the inequality is in standard form, you have two options to
determine on which side to shade1) write the inequality in
slope-intercept form and follow the rules given2) pick a test point
that is not on the line and see if the inequality is true at that
pointShade on the side of the line which is trueYou are only
allowed one graph per coordinate plane when dealing with
inequalitiesEx1. Is (3, -6) a solution to y < -3x + 5?Graph each
inequality on its own graph
Ex2.
Ex3. 3x 4y > 12
Ex4. 8x + 6y > 24
7-6 Systems of Linear InequalitiesCombining two or more linear
inequalities creates a system of linear inequalitiesThe solution is
the area where all of the inequalities are true (where the shading
overlaps)Do all of the original shading VERY lightly and then
darken in the final answerFor a test point to be a solution to the
system, it must work for all of the inequalitiesOne question per
coordinate gridEx1. Is (3, 5) a solution to the following
system?
Solve each system by graphingEx2.
Ex3.
Ex3. Suppose you have two jobs, babysitting, which pays $5 per
hour, and sacking groceries, which pays $6 per hour. You can work
no more than 20 hours each work, but you need to earn at least $90
per week. How many hours can you work at each job? Write a system
of inequalities and graph.
