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Systems of Equations & Inequalities
Is the given value a solution?
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Stacking Cups (101Qs) & Oreos (Desmos)
Find the solution by looking at graphs
This is probably the easiest topic we'll cover all year! :-)
When a system of equations is graphed, the solution is the point where the two lines intersect.
That's literally it.
When a system of equations is graphed, the solution is the point where the two lines intersect.
That's literally it.
Number your paper from 1 to 18. Every answer should be in the form (x,y).
systems_graphs_setof18.pdf |
Keep in mind...there are some special cases when the lines don't intersect at exactly one point...
Solve a system by substitution (with an isolated variable)
Click HERE to check out Cool Math's description of solving systems by substitution. Their explanations are REALLY good, and they color code! :-)
*** Start at 1:39 and watch until 3:25
Solve a system by substitution AKA f(x) = g(x) or Eq1 = Eq2
*** Start at 2:13
fxgx.pdf |
Solve a system by elimination
We call it "elimination"...they call it "linear combination"...same thing!
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Elimination: Practice Problems (With Answers!)
Only focus on Problems 1-16. The solutions are on the 3rd and 4th pages.
# 1-4 will cancel as is, no transformations are needed.
# 5-8 require a flip the signs transformation
# 9-16 require a complete transformation, either positive or negative
# 1-4 will cancel as is, no transformations are needed.
# 5-8 require a flip the signs transformation
# 9-16 require a complete transformation, either positive or negative
systems_elimination_ans.pdf |
Worked-Through Examples
Graphing Inequalities
There are 2 things that you must understand about graphing inequalities.
(1) SOLID VS. DASHED LINE
If an inequality has ≤ or ≥, then the line will be SOLID. (This is analogous to a closed point.)
If an inequality has < or >, then the line will be DASHED. (This is analogous to an open point.)
(2) SHADE ABOVE VS. BELOW
In an inequality has > or ≥, then you will shade ABOVE the line. (If the line is a vertical x=# line, then shade to the right.)
If an inequality has < or ≤, then you will shade BELOW the line. (If the line is a vertical x=# line, then shade to the left.)
Even though the image below has non-linear quadratic functions, make note of the inequality symbol used, the type of line (solid vs. dashed), and where the graph is shaded (above vs. below).
(1) SOLID VS. DASHED LINE
If an inequality has ≤ or ≥, then the line will be SOLID. (This is analogous to a closed point.)
If an inequality has < or >, then the line will be DASHED. (This is analogous to an open point.)
(2) SHADE ABOVE VS. BELOW
In an inequality has > or ≥, then you will shade ABOVE the line. (If the line is a vertical x=# line, then shade to the right.)
If an inequality has < or ≤, then you will shade BELOW the line. (If the line is a vertical x=# line, then shade to the left.)
Even though the image below has non-linear quadratic functions, make note of the inequality symbol used, the type of line (solid vs. dashed), and where the graph is shaded (above vs. below).
Systems of Inequalities
The red line has a ≤ symbol, so you make the line SOLID and shade red UNDER the line.
The blue line has a > sumbol, so you make the line DASHED and shade blue ABOVE the line.
The solution is the PURPLE area, because that is where you shaded BOTH the red and blue inequalities.
A way to check is to take an "easy" point (with 0's or small numbers) and plug it into both inequalities.
If you plug (0,0) into the red inequality, you end up with 0 ≤ -2. This is FALSE, which is why (0,0) is not shaded in red.
If you plug (0,0) into the blue inequality, you end up with 0 > 5. This is FALSE, which is why (0,0) is not shaded in blue.
Because (0,0) is not a solution for either inequality, it is not shaded in any color.
If you plug (5,0) into the red inequality, you end up with 0 ≤ 3. This is TRUE, which is why (5,0) is shaded in red.
If you plug (5,0) into the blue inequality, you end up with 0 > -10. This is TRUE, which is why (5,0) is shaded in blue.
Because (5,0) is a solution for both inequalities, it is shaded in purple (where purple represented the overlap of red and blue).
If you plug (2,2) into the red inequality, you end up with 2 ≤ 0. This is FALSE, which is why (2,2) is not shaded in red.
If you plug (2,2) into the blue inequality, you end up with 2>1. This is TRUE, which is why (2,2) is shaded in blue.
Because (2,2) is NOT a solution for both inequalities, it is not shaded in purple.