3.1 Solving Systems of Linear Equations Algebraically

3.1 Solving Systems of Linear Equations Algebraically

Quick Summary: A system of two linear equations consists of two equations with two shared variables (typically $x$ and $y$). Solving the system algebraically means finding the ordered pair $(x, y)$ that satisfies both equations simultaneously. On the Digital SAT, mastering both substitution and elimination, knowing when to use each, and recognizing shortcuts for compound expressions (such as $x + y$ or $2x - y$) will save valuable test time.

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The Anatomy of a 2x2 Linear System

A two-variable linear system typically appears in standard form or slope-intercept form:

$\begin{cases} A_1x + B_1y = C_1 \\ A_2x + B_2y = C_2 \end{cases}$

A solution is an ordered pair $(x_0, y_0)$ that produces a true statement when substituted into both equations. If an ordered pair satisfies one equation but not the other, it is not a solution to the system.

                  +-------------------------------+
                  |    2x2 Linear System          |
                  |  { A_1x + B_1y = C_1          |
                  |  { A_2x + B_2y = C_2          |
                  +---------------+---------------+
                                  |
         +------------------------+------------------------+
         |                                                 |
         v                                                 v
+------------------------------+                  +------------------------------+
|     Substitution Method      |                  |      Elimination Method      |
|  Best when one coefficient   |                  |   Best when coefficients     |
|  is 1 or -1 (e.g., y = 2x-5) |                  |   align nicely (e.g., 3x/-3x)|
+------------------------------+                  +------------------------------+

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Method 1: The Substitution Method

The substitution method works by isolating one variable in one equation and substituting its equivalent expression into the other equation. This transforms a two-variable system into a single-variable linear equation.

Step-by-Step Substitution Protocol

1. Isolate: Choose the variable with the simplest coefficient (ideally $+1$ or $-1$) and isolate it on one side of an equation.
2. Substitute: Replace that variable in the other equation with the expression found in Step 1.
3. Solve: Solve the resulting single-variable equation.
4. Back-Substitute: Plug the numerical solution back into your isolated variable expression to find the second variable.
5. Check: Verify the ordered pair $(x, y)$ in the original equations or in Desmos.

Worked Example 1: Substitution

$\begin{cases} 2x + y = 11 \\ 3x - 4y = 11 \end{cases}$

• Step 1: Isolate $y$ in the first equation. $y = 11 - 2x$

• Step 2: Substitute $(11 - 2x)$ for $y$ in the second equation. $3x - 4(11 - 2x) = 11$

• Step 3: Distribute and solve for $x$. $3x - 44 + 8x = 11$ $11x - 44 = 11$ $11x = 55 \implies x = 5$

• Step 4: Back-substitute $x = 5$ into the isolated expression. $y = 11 - 2(5) = 11 - 10 = 1$

The solution is $(5, 1)$.

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Method 2: The Elimination Method (Linear Combinations)

The elimination method involves multiplying one or both equations by non-zero constants so that the coefficients of one variable become exact opposites (e.g., $+4y$ and $-4y$). Adding the two equations vertically eliminates that variable.

Step-by-Step Elimination Protocol

1. Align: Write both equations in standard form ($Ax + By = C$) with like terms aligned vertically.
2. Scale: Multiply one or both equations by suitable non-zero constants to make the coefficients of one chosen variable additive inverses (opposite signs, same absolute value).
3. Add: Add the equations vertically to eliminate one variable.
4. Solve: Solve the remaining one-variable equation.
5. Back-Substitute: Plug the value into either original equation to determine the other variable.

Worked Example 2: Elimination

$\begin{cases} 3x + 2y = 16 \\ 5x - 3y = 14 \end{cases}$

• Step 1: Choose a variable to eliminate. To eliminate $y$, multiply the first equation by $3$ and the second equation by $2$: $3 \cdot (3x + 2y = 16) \implies 9x + 6y = 48$ $2 \cdot (5x - 3y = 14) \implies 10x - 6y = 28$

• Step 2: Add the two equations together. $(9x + 6y) + (10x - 6y) = 48 + 28$ $19x = 76 \implies x = 4$

• Step 3: Back-substitute $x = 4$ into the first original equation. $3(4) + 2y = 16$ $12 + 2y = 16 \implies 2y = 4 \implies y = 2$

The solution is $(4, 2)$.

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Method Selection Matrix

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High-Yield Digital SAT Shortcut: Direct Linear Combinations

A classic Digital SAT test design pattern asks students to find a compound value such as $x + y$, $x - y$, $3x + 3y$, or $7x + 7y$, rather than $x$ or $y$ alone. Unprepared students solve for both variables individually and add them, wasting 30 to 45 seconds.

The Linear Combination Shortcut

Always inspect the sum and difference of the two given equations. Often, adding or subtracting the equations directly yields the target compound expression or an integer multiple of it.

Worked Example 3: Finding a Compound Expression in One Step

$\begin{cases} 4x + 3y = 17 \\ 2x + 3y = 11 \end{cases}$ Question: What is the value of $2x$?

• Traditional Method: Eliminate $y$ to get $x = 3$, back-substitute into $4x + 3y = 17$ to find $3y = 5 \implies y = \frac{5}{3}$, then finally compute $2x = 2(3) = 6$.
• SAT Shortcut: Subtract the second equation from the first equation directly: $(4x + 3y) - (2x + 3y) = 17 - 11$ $2x = 6$

The target quantity $2x$ is found in one step without ever calculating $x$ or $y$.

Worked Example 4: Scaling to a Compound Sum

$\begin{cases} 5x + 2y = 23 \\ 3x + 6y = 33 \end{cases}$ Question: What is the value of $x + y$?

• Solution: Add the two equations directly: $(5x + 2y) + (3x + 6y) = 23 + 33$ $8x + 8y = 56$
• Factor out $8$ on the left side: $8(x + y) = 56$ $x + y = \frac{56}{8} = 7$

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Handling Fractions and Decimals in Systems

When a system contains fractions or decimals, eliminate them in Step 0 before performing substitution or elimination:

• Fractions: Multiply the entire equation by the Least Common Denominator (LCD) of all terms.
• Decimals: Multiply by $10, 100,$ or $1000$ to shift decimal points to whole numbers.

$\begin{cases} \frac{1}{3}x - \frac{1}{2}y = 1 \\ 0.2x + 0.4y = 2.8 \end{cases}$

1. Multiply Eq 1 by $\text{LCD} = 6$: $6\left(\frac{1}{3}x - \frac{1}{2}y\right) = 6(1) \implies 2x - 3y = 6$.
2. Multiply Eq 2 by $10$: $10(0.2x + 0.4y) = 10(2.8) \implies 2x + 4y = 28$.
3. Subtract Eq 1 from Eq 2: $(2x + 4y) - (2x - 3y) = 28 - 6 \implies 7y = 22 \implies y = \frac{22}{7}$
4. Substitute $y = \frac{22}{7}$ into $2x - 3y = 6$: $2x - 3\left(\frac{22}{7}\right) = 6 \implies 2x = 6 + \frac{66}{7} = \frac{108}{7} \implies x = \frac{54}{7}$

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Non-Standard Forms and Parentheses Expansion

Often on harder SAT modules, equations are presented with unexpanded parentheses or variables scattered across both sides:

$3(x - 2) + 4(y + 1) = 2x + 15$

1. Distribute completely: $3x - 6 + 4y + 4 = 2x + 15 \implies 3x + 4y - 2 = 2x + 15$.
2. Rearrange to Standard Form ($Ax + By = C$): $3x - 2x + 4y = 15 + 2 \implies x + 4y = 17$
3. Now the equation is ready for rapid substitution ($x = 17 - 4y$) or elimination.

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Digital SAT Strategy: Desmos Verification Workflow

1. Type Equations As Given: In the built-in Desmos calculator, you do not need to convert equations to $y = mx + b$. Type $4x + 3y = 25$ directly into Line 1 and $2x - 3y = -7$ into Line 2.
2. Click the Intersection Point: Desmos automatically places a gray dot at the intersection. Click the dot to display $(x, y)$.
3. Re-Read the Prompt: Ensure you answer the exact question asked (e.g., $x$, $y$, $x+y$, or $xy$).

+-----------------------------------------------------------------------------+
|                      DESMOS SYSTEM VERIFICATION GUIDE                       |
|                                                                             |
|   Line 1: 4x + 3y = 25                                                      |
|   Line 2: 2x - 3y = -7                                                      |
|                                                                             |
|   Visual Canvas:                                                            |
|   - Both lines are drawn automatically.                                     |
|   - Gray dot appears at (3, 4.333).                                         |
|   - Click the dot: Coordinates (3, 13/3) lock onto the graph.               |
|                                                                             |
|   Time to solve: ~10 seconds.                                               |
+-----------------------------------------------------------------------------+

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Common Traps to Avoid

• Answering for the wrong variable: You solved for $x = 3$, but the question asked for $y$ or $x - y$. Always re-read the final sentence of the prompt before selecting your answer.
• Sign errors during subtraction: When subtracting $(2x - 3y)$, remember that $-(-3y) = +3y$. Distribute the negative sign across all terms.
• Forgetting to multiply the constant term on the RHS: When scaling an equation like $2x + 3y = 12$ by $3$, remember to multiply the right side: $6x + 9y = 36$, not $12$.
• Premature rounding on fractional SPRs: Never convert intermediate fractions to short decimals (e.g., $13/3$ to $4.33$). Keep exact fractions until the final step.