AMC 10 · 2006 · #12
Grade 8 algebraPick an answer.
AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The point (1,2) is where the lines meet, which means it sits on each line — so its coordinates must satisfy both equations. Tool #4 (Introduce a Variable) is really about setting up and solving equations for an unknown: here a and b are the unknowns, and substituting x=1, y=2 turns each two-variable line into a one-line equation that hands over a (from the first) and b (from the second). Add them for a+b. Tool #15 (Organize Information in More Ways) offers a slicker route — add the two original equations before substituting so a+b appears together in one step. Tool #3 (Eliminate Possibilities) then confirms the result against the five listed choices.
Use that the point lies on both lines
The lines meet at (1,2), so that one point sits on both — meaning x=1 and y=2 must satisfy each equation.
An intersection point is the one place both lines agree, so it obeys both rules at once.
An intersection point is the one place both lines agree, so it obeys both rules at once.
▸ Why?
A point on a line satisfies that line's equation, and this point lies on both lines.
▸ Why?
Substituting the known coordinates keeps each equation true and leaves only the unknown letter.
Plug the point into the first line to get a
Put the point into x=(1/4)y+a: 1=(1/4)·2+a=1/2+a, so a=1/2.
With the point substituted, the only thing left to balance the equation is a itself.
6.EE.B.7Introduce A VariablePlug the point into the second line to get b
Put the same point into y=(1/4)x+b: 2=(1/4)·1+b=1/4+b, so b=8/4-1/4=7/4.
The same substitution trick empties the second equation of everything but b.
6.EE.B.7Introduce A VariableAdd a and b
Match denominators and add: a+b=1/2+7/4=2/4+7/4=9/4, which is choice (E).
To add fractions you first make the bottoms match, then just add the tops.
5.NF.A.1Introduce A VariableAn intersection point sits on both lines, so plug its coordinates into each equation — everything else drops out and the unknown constants fall right into your hands.
- Use that the point lies on both lines
- Plug the point into the first line to get a
- Plug the point into the second line to get b
- Add a and b