Direct Calculation

This pattern asks you to solve a system of two linear equations to find the values of the variables (or an expression involving them). The two main methods are substitution and elimination.

 

Substitution: One Variable Already Isolated

When one equation is already solved for a variable (like $c = 9d$), substitute that expression into the other equation.

$c + d = 30$
$c = 9d$

What is the solution $(c, d)$?

A) $(27, 3)$
B) $(20, 10)$
C) $(18, 12)$
D) $(25, 5)$

Substitute $c = 9d$ into the first equation: $9d + d = 30$, so $10d = 30$, giving $d = 3$. Then $c = 9(3) = 27$. The answer is A.

$y = -2x$
$x - 3y = 21$

What is the value of $x$?

A) $-6$
B) $3$
C) $7$
D) $21$

Substitute $y = -2x$ into the second equation: $x - 3(-2x) = 21$, so $x + 6x = 21$, giving $7x = 21$ and $x = 3$. The answer is B. Note that $-6$ is the value of $y$, not $x$.

$a = 3b$
$4a + 2b = 28$

What is the value of $b$?

A) 2
B) 6
C) 7
D) 14

Substitute $a = 3b$: $4(3b) + 2b = 28$, so $12b + 2b = 28$, giving $14b = 28$ and $b = 2$. The answer is A. Option B (6) is the value of $a$, not $b$.

 

Substitution: Isolate First, Then Substitute

When neither variable is isolated, solve one equation for a variable and substitute.

$y = 2x + 1$
$3x + 2y = 16$

What is the value of $x$?

Substitute $y = 2x + 1$ into the second equation: $3x + 2(2x + 1) = 16$, so $3x + 4x + 2 = 16$, giving $7x = 14$ and $x = 2$.

$x = 4y - 2$
$3x - y = 5$

What is the value of $y$?

Substitute $x = 4y - 2$ into the second equation: $3(4y - 2) - y = 5$, so $12y - 6 - y = 5$, giving $11y = 11$ and $y = 1$.

 

Elimination: Matching Coefficients

When both equations are in standard form ($ax + by = c$), you can add or subtract the equations to eliminate one variable.

$7k - j = 8$
$2k + j = 10$

What is $(k, j)$?

Add the two equations to eliminate $j$: $(7k - j) + (2k + j) = 8 + 10$, so $9k = 18$ and $k = 2$. Substitute back: $2(2) + j = 10$, so $j = 6$. The solution is $(2, 6)$.

The coefficients of $j$ are $-1$ and $+1$, which cancel when you add. If they don't cancel, multiply one or both equations first.

$3a - 4b = 7$
$3a - 2b = 11$

What is the value of $b$?

Subtract the second from the first: $(3a - 4b) - (3a - 2b) = 7 - 11$, so $-2b = -4$ and $b = 2$.

$5c - 8d = 11$
$-5c + 3d = -1$

What is $(c, d)$?

Add the equations: $(5c - 8d) + (-5c + 3d) = 11 + (-1)$, so $-5d = 10$ and $d = -2$. Substitute: $5c - 8(-2) = 11$, giving $5c + 16 = 11$, so $5c = -5$ and $c = -1$. The solution is $(-1, -2)$.

 

Elimination with Multipliers

When no coefficients match or cancel, multiply one or both equations to create matching coefficients.

$4x - 20 = 3y$
$4x - 5y = 40$

Rewrite the first equation: $4x - 3y = 20$. Now the system is: $4x - 3y = 20$
$4x - 5y = 40$

Subtract: $(4x - 3y) - (4x - 5y) = 20 - 40$, so $2y = -20$ and $y = -10$. Then $4x = 20 + 3(-10) = -10$, so $x = -2.5$.

$6u + 5v = 15$
$2u - 3v = 13$

What is the value of $u$?

Multiply the second equation by 3: $6u - 9v = 39$. Now subtract from the first: $(6u + 5v) - (6u - 9v) = 15 - 39$, so $14v = -24$ and $v = -\dfrac{12}{7}$.

Alternatively, multiply the second by $-3$ and add to eliminate $u$ directly.

 

Systems with Parametric Coefficients

Hard questions may include constants as coefficients ($m$, $n$) and give additional information to find them.

$mx + ny = 10$
$nx + my = 8$

If the solution is $(x, y) = (2, 1)$, what is $m + n$?

Substitute $(2, 1)$: $2m + n = 10$ and $2n + m = 8$. This is a new system! Add: $3m + 3n = 18$, so $m + n = 6$.

 

Finding an Expression (Not Individual Variables)

Sometimes the question asks for $a + b$ or $v - u$ rather than individual values. Look for shortcuts — adding or subtracting the equations may directly give the target expression.

$4a = -8$
$-3a + b = 12$

What is $a + b$?

A) 4
B) $-4$
C) 20
D) $-20$

Shortcut: Add the two equations. $(4a) + (-3a + b) = -8 + 12$, so $a + b = 4$. The answer is A. No need to solve for each variable separately.

$6u = 12$
$v - 7u = -4$

What is $v - u$?

Add: $(6u) + (v - 7u) = 12 + (-4)$, so $v - u = 8$. Done in one step.

When the target expression doesn't appear from simply adding or subtracting, solve for the individual variables first, then compute the expression.