Depth · one concept from first principles · step 2 of 4
A zero product graphs as two lines
Count the points (x, y) that satisfy both (y-2)(x-2y+2)=0 and (2x+y-5)(x-2y-2)=0.
Expressions
Your answer
How to solve
Strategy Organize Information in More Ways — Re-read each equation: a product is zero when one factor is zero, so each equation's graph is a PAIR of lines. A common solution is a crossing between a line from the first pair and a line from the second, so list the four pairings and check which actually meet.
1STEP 1
Split each equation into two lines
Equation 1 is y=2 or y=(1/2)x+1; equation 2 is y=-2x+5 or y=(1/2)x-1.
Each factored equation draws two lines, not one curve.
8.EE.C.8Organize Information In More WaysEach factored equation is a pair of lines, so a common solution is a crossing of a line from the first pair with a line from the second.
Why?
A product is zero only when one of its factors is zero, so means or , and likewise for the second equation.
🧱A product is zero only if a factor is zeroIf two numbers multiply to zero then at least one of them was zero, so a factored equation hands you one case per factor.
Why?
A point solves both equations exactly when it makes one factor of each equation true, which is the same as sitting on one line from each pair.
🧱A solution makes the equation trueA solution is a number that, put in for the unknown, makes both sides come out equal; that is all being a solution means.
2STEP 2
Pair 1 with 1, and 1 with 2
y=2 with y=-2x+5: 2=-2x+5, so (3/2, 2). y=2 with y=(1/2)x-1: 2=(1/2)x-1, so (6, 2).
Each crossing is a two-equation system with one solution.
8.EE.C.8Make A Systematic List3STEP 3
The other two pairs
y=(1/2)x+1 meets y=-2x+5 at (8/5, 9/5); it is parallel to y=(1/2)x-1 (same slope), so that pair never meets.
Parallel lines contribute nothing.
8.EE.C.8Make A Systematic List4STEP 4
Count distinct points
The three points (3/2, 2), (6, 2), (8/5, 9/5) are all different, so there are 3 ordered pairs.
1+1+1+0=3
Check for repeats: a point counted twice would make the total smaller.
8.EE.C.8Draw A DiagramAnswer
3
Plug in (8/5, 9/5): 8/5-18/5+2=0 makes equation 1 zero, and 16/5+9/5-5=0 makes equation 2 zero. The other two points sit on y=2 and on one line of equation 2 by construction.
Takeaway
A product equal to zero is really two simpler equations wearing one coat.
- Split each equation into two lines
- Pair 1 with 1, and 1 with 2
- The other two pairs
- Count distinct points
Where next?
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