AIMS Practice Session – Notes
Solving with variables on both sides
1st step – Distribute to get rid of the ( )
2nd step – combine like terms (only on same side of equal sign)
3rd step – get variables on one side of the equal sign and numbers on the other side using inverse operations
4th step – continue to undo operations to isolate the variable
5th step – check by plugging your answer back into the original problem
1.
k + 7 = 2k + 1
3. –3(3x + 15) – (10 + x) = 35
2. 6(5k – 8) – 20 = 11(2k – 3) + 3k
4. –2(n + 15) + 3 = –5 –2 (2n +10)
Solving Proportions
Reminder: To solve a proportion, you cross multiply, which will get rid of the fractions. If needed, do the distributive property.
Example:
1.
3
2

would cross multiply as 3x  5  2  8 , not 3x  5  2  8 .
8 x5
3 x

4 12
2.
8 3x  4

5
10
3.
r 3 5

r 9 8
Solving Inequalities
Treat these the same as the equations we solved earlier – get the variables on one side and the numbers on the other. There is only
one small thing to remember here – if you multiply or divide by a negative value, you need to switch the direction of the inequality.
When graphing, < or > is an open circle, and  or  is a closed circle.
1.
 8  6x  104
2.
2  3x  8 x  4(6  2 x)
3.
7  2n  71  6n  7n
Graphing Inequalities
Things to remember about graphing inequalities:
y  mx  b form. M is your slope and b is the y-intercept.
1)
Before you graph, make sure you get the equation into
2)
3)
4)
Remember – multiplying or dividing by a negative means you switch the inequality.
Make a solid line if the inequality is  or  ; make a dashed line if the inequality is < or >
To figure out which way to shade,: if the sign is  or <, shade down (where the y’s are less than the line). If the sign is  or
>,shade up (where the y’s are greater than the line).
1.
Graph
1
y   3
4
2.
Graph
9 x  3 y  3
3.
Graph
3 x  5 y  20
Systems of Equations – Parallel, perpendicular, coinciding or intersecting but not perpendicular
1)
2)
The equations must be in y=mx+b form to compare slopes.
Parallel lines have the same slope. These lines will also have no solution if you try to solve for their intersection (you may
get an answer that looks like 2=6 or some other statement that can never be true).
3)
Perpendicular lines have slopes that are flipped and opposite signs (like 4 and
1
 ). The point of intersection is the
4
6)
solution.
Lines that are coinciding are exactly the same line – one is just a multiple of the other. You will get an answer of 2=2 or a
statement that will always be true.
If you solve a system of equations and you get one and only one answer, but the slopes are not perpendicular, the lines are
intersecting but not perpendicular.
You can solve systems of equations by substitution, elimination, graphing, or comparing slopes.
1.
Tell whether the lines are parallel, perpendicular, coinciding, or intersecting but not perpendicular.
4)
5)
3x  4 y  4
4 x  3 y  12
2.
Tell whether the lines are parallel, perpendicular, coinciding, or intersecting but not perpendicular.
2x  5 y  6
 6 x  15 y  18
3.
Tell whether the lines are parallel, perpendicular, coinciding, or intersecting but not perpendicular.
 4x  2 y  8
 y  6  2x
4.
Tell whether the lines are parallel, perpendicular, coinciding, or intersecting but not perpendicular.
y  3  2x
x  4 y  3