Systems of Linear Equations

Pentru elevii români: Metodele sunt aceleași ca la școală (substituția și metoda reducerii, numită în engleză elimination), dar SAT-ul întreabă des câte soluții are un sistem judecând după pante și intercepturi, nu cu determinanți. În plus, ai un calculator Desmos integrat în care poți vedea direct punctul de intersecție al dreptelor.

What the SAT tests here

Systems of two linear equations in two variables belong to the Algebra domain. Expect three kinds of questions:

1. Solve the system, often for just one variable or for an expression such as $x + y$.
2. Count the solutions: one, none, or infinitely many, frequently with an unknown constant.
3. Model a situation (tickets, mixtures, coins, prices) with two equations and solve.

Key rules and formulas

A solution $(x, y)$ is a point that satisfies both equations: the intersection point of two lines.

Substitution: solve one equation for a variable and substitute into the other. Best when a variable already has coefficient 1, as in $y = 2x - 1$.

Elimination: multiply one or both equations so that one variable has opposite coefficients, then add the equations.

Number of solutions. For $ax + by = c$ and $dx + ey = f$:

In slope-intercept form: different slopes give one solution; same slope with different intercepts gives none; same slope with the same intercept gives infinitely many. (If any coefficient in the denominators is 0, compare slopes instead of ratios.)

Worked examples

Example 1 (elimination). Solve $3x + 2y = 16$ and $5x - 2y = 0$.

The $y$-terms are already opposites, so add the equations: $8x = 16$, so $x = 2$. Substitute into the first equation: $6 + 2y = 16$, so $y = 5$. Check in the second equation: $5(2) - 2(5) = 0$. ✓

Example 2 (word problem). A school theater sold 150 tickets for a total of $1,450. Adult tickets cost $12 and student tickets cost $7. How many adult tickets were sold?

Let $a$ = adult tickets and $s$ = student tickets.

$a + s = 150 \qquad 12a + 7s = 1{,}450$

Substitute $s = 150 - a$: $12a + 7(150 - a) = 1{,}450$, so $5a + 1{,}050 = 1{,}450$ and $a = 80$. Then $s = 70$. Check: $12(80) + 7(70) = 960 + 490 = 1{,}450$. ✓

Example 3 (no solution). In the system $kx + 6y = 9$ and $2x + 3y = 5$, $k$ is a constant. For what value of $k$ does the system have no solution?

For parallel lines, the coefficients of $x$ and $y$ must be in the same ratio: $\frac{k}{2} = \frac{6}{3} = 2$, so $k = 4$. Check the constants: $\frac{9}{5} \neq 2$, so the lines are parallel and distinct, not identical. The answer is $k = 4$.

Test-taking strategy

• Read what is being asked. If the question asks for $x + y$ or $x - y$, look for a shortcut before solving completely. For example, with $3x + 5y = 31$ and $5x + 3y = 33$, adding gives $8x + 8y = 64$, so $x + y = 8$ immediately.
• Desmos for solving. Type each equation exactly as given (Desmos accepts 3x+2y=16 without rearranging). Click the intersection point to see its coordinates. If the coordinates are long decimals and the answers are fractions, compare them by typing the fractions into Desmos too.
• Desmos for constants. Type kx+6y=9 and add the slider for $k$. Watch the lines: the moment they become parallel, you have the "no solution" value. Confirm it algebraically, because a slider moves in steps and can skip the exact value.
• Define variables in writing. In word problems, write "$a$ = adult tickets" before setting up equations. Many wrong answer choices are the other variable.
• Check with the original words, not your equations; a wrong equation can give a "solution" that still checks in itself.

Common traps

• Answering for the wrong variable. Trap choices often equal the other unknown (here, 70 student tickets instead of 80 adult tickets).
• Confusing "no solution" with "infinitely many." Parallel lines (same slope, different intercepts) never meet; identical lines meet everywhere. Always check the constants.
• Sign errors in elimination. When you subtract equations, subtract every term, including the constant.
• Forgetting to multiply the constant when you scale an equation: $2(x + 3y = 4)$ is $2x + 6y = 8$, not $2x + 6y = 4$.
• Assuming the intersection must be an integer. SAT systems can have fractional solutions; trust your work and verify.

Summary

A system's solution is the intersection of two lines. Use substitution when a variable is already isolated and elimination when coefficients line up. Count solutions by comparing slopes and intercepts or coefficient ratios. In word problems, define your variables, build two equations, and answer the exact quantity asked. Desmos shows the intersection instantly and lets you test constants with sliders.