Systems of two linear equations
A system asks for the point that satisfies both equations at once — where the two lines cross. You'll solve them by substitution or elimination, and judge whether a system has one solution, none, or infinitely many.
What College Board tests
Solving a system for its single solution, setting up a system from a word problem, and determining the number of solutions by comparing the lines' slopes and intercepts.
Two methods
How many solutions
Compare the two lines. Different slopes → they cross once → one solution. Same slope, different intercepts → parallel, never meet → no solution. Same slope and same intercept → the same line → infinitely many solutions.
Four worked examples in SAT format. Read the approach, try it yourself, then tap Show the full solution.
1 · Solve by substitution
For the system below, what is the value of ?
Approach The first equation already gives by itself. Substitute that expression for in the second equation, so only is left.
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Answer: B
Why the other choices are wrong
A Forgets the when substituting.
C This is the value of from a sign error; check by plugging back in.
D This is , not .
2 · Solve by elimination
For the system below, what is the value of ?
Approach Both equations start with . Subtract one equation from the other and the -terms cancel, leaving only .
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Answer: B
Why the other choices are wrong
A Mishandles the double negative .
C This is the value of , not .
D Divides 8 by the wrong number.
3 · A system with no solution
How many solutions does the following system have?
- Exactly one
- Exactly two
- Zero
- Infinitely many
Approach Check whether the two lines are really the same line, parallel, or crossing. A quick test: see if one equation is a multiple of the other. If the variable sides match up but the constants don't, the lines are parallel — no solution.
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Answer: C
Why the other choices are wrong
A Would require different slopes; here the slopes are equal.
B Two distinct lines can cross at most once, never twice.
D Would require the constants to match too (both 20); they don't.
4 · Set up a system from a situation
A concession stand sells small drinks for $2 and large drinks for $3. In one hour it sold 50 drinks for a total of $130. If is the number of small drinks and the number of large drinks, which system represents this situation?
Approach Write one equation for the count of drinks and one for the money. Keep prices attached to the right drink: $2 goes with small, $3 with large.
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Answer: B
The prices are rates. "$3 per large" means "$3 for each one large drink," so written as a fraction its denominator is 1 large drink:
Multiplying a price by a count of drinks cancels the "drink" unit and leaves dollars, so every term in the money equation comes out in dollars:
The same holds for the $2 small drinks, so the two terms add to a total in dollars — matching the $130.
Why the other choices are wrong
A Swaps the totals — 50 is the count, 130 is the money.
C Attaches the prices to the wrong drinks ($3 to small).
D Swaps which equation gets which total.