Mistake Master
Solving systems and reading solutions
A system of linear equations is two lines at once, and its solution is where they meet. The math is quick: substitute one equation into the other, or line them up and eliminate a variable. The points leak in three places, a sign dropped during substitution, a no-solution or all-solutions result read backward, and the right value reported for the wrong quantity. Carry every sign, read what the canceled variables say, then answer exactly what the item asks.
§1
What this topic is about
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A system of linear equations is two equations that have to hold at the same time. On the coordinate plane they are two lines, and the solution is the point where the lines cross. The work splits in two: solve the system, by substitution or by elimination, then report the exact quantity the question names. The arithmetic is short. The points leak in a dropped sign, in misreading what happens when the variables cancel, and in answering for one variable when the item wanted a combination.
§2
Substitution: replace, then keep every sign
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When one equation already gives a variable on its own, like $y = 2x - 1$, substitution is fastest: put that expression in for the variable in the other equation, then solve the single equation that results. The one move that costs points is letting a substituted negative term turn positive while you combine. The term keeps the sign it had.
- Substitute inside parentheses so the sign travels with the term: $3x + (2x - 1) = 9$.
- Combine without flipping: that is $5x - 1 = 9$, not $5x + 1 = 9$. The $-1$ stays negative.
- Solve for the first variable, then back-substitute to get the second. Here $x = 2$, then $y = 2(2) - 1 = 3$.
Worked example. Solve $y = 2x - 1$ and $3x + y = 9$ for $x$.
Replace $y$ in the second equation: $3x + (2x - 1) = 9$.
$$5x - 1 = 9 \implies 5x = 10 \implies x = 2.$$ Turning the $-1$ into $+1$ gives $5x + 1 = 9$ and $x = \dfrac{8}{5}$, a wrong answer built entirely on the dropped sign.
§3
Elimination: line them up and subtract the whole equation
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When both equations are in the same $ax + by = c$ form, elimination is fastest: add or subtract them so one variable cancels. The slip here is subtracting the left sides but leaving the right side whole, which changes one side of the equation and not the other. Whatever you do to one side, do to both.
- Match a variable's coefficient, then subtract every term on both sides, constants included.
- If coefficients differ, scale a whole equation first, both sides, so the variable lines up to cancel.
- Once one variable is found, put it back into either original equation to get the other.
Worked example. Solve $2x + y = 11$ and $x + y = 7$ for $x$.
$(2x + y) - (x + y) = 11 - 7$, so $x = 4$.
$$4 + y = 7 \implies y = 3.$$ Subtracting the left sides but keeping $11$ on the right gives $x = 11$, the same equation with one side changed and the other left alone.
§4
How many solutions: one, none, or infinitely many
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A system of two lines has exactly one of three outcomes, and they are easy to mix up. Solve until the variables cancel, then read the leftover statement. A true statement like $0 = 0$ means the equations describe the same line; a false one like $0 = 5$ means the lines never meet. Anything else means a single crossing.
- One solution: different slopes, so the lines cross once. This is the usual case.
- No solution: equal slopes, different intercepts. Parallel lines, never meeting. The variables cancel to a false statement.
- Infinitely many: one equation is a multiple of the other, constants included, so they are the same line. The variables cancel to a true statement.
Worked example. Two lines are graphed below. How many solutions does the system have?
The lines have different slopes and cross once, so there is exactly one solution. Reading parallel lines as crossing, or a repeated line as crossing once, is how the other two counts get chosen by mistake. A pair of straight lines can never share exactly two points.
§5
Report the exact quantity the item asks for
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Solving the system is usually the easy part. The trap is the last step: the item asks for $x + y$, you found $x = 4$, and you stop there. The number is real, it just answers a question that was not asked. Before solving, underline what the item wants, then read that off the finished solution.
Worked example. A system solves to $x = 4$ and $y = 1$. The item asks for $x + y$. What is the answer?
The system gives $x = 4$ and $y = 1$.
$$x + y = 4 + 1 = 5.$$ Stopping at $x = 4$ answers a different question; reporting $y = 1$ names the wrong variable. The item asked for the sum, so the answer is $5$.
Keep every sign through the substitution, read the leftover statement to count solutions, and report the exact quantity the item names, not one variable when it asked for a combination.
§6
Three patterns that cost real points
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Three patterns recur on systems. They are the same ones the diagnostic routes on.
A sign drops during substitution.
The substituted term carries a minus, and it turns positive while combining: $3x + (2x - 1) = 9$ gets written as $5x + 1 = 9$. The arithmetic after that is clean, but it started from the wrong sign, so the solution drifts.
Fix. Substitute inside parentheses and keep the sign attached. A $-1$ stays $-1$ after combining.
A no-solution or all-solutions result is read backward.
Parallel lines get called one solution, a repeated line gets called one solution, or matching coefficients get read as parallel when the constants match too and the lines are actually identical.
Fix. Cancel the variables and read the leftover: $0 = 5$ is no solution, $0 = 0$ is infinitely many, anything else is one crossing.
One variable is reported when a combination was asked.
The system is solved correctly, then the answer is $x$ when the item wanted $x + y$, or the unasked variable. The number is right for a question the item never asked.
Fix. Underline what the item asks for before solving, then read that exact quantity off the finished solution.
Ten quick checks across the three patterns: keeping the sign through a substitution, reading one, none, or infinitely many solutions, and reporting the exact quantity asked. Pick or type your answer, then check. Progress is saved.