UNIT 7 • ALGEBRA

System of Equations
Cramer, Elimination, Substitution

A system of linear equations asks for values that satisfy every equation at once. With 2 equations, that is where two lines cross. Cramer's rule gives each unknown as a ratio of determinants, $x = \frac{D_x}{D}$ .

CRAMER'S RULE (2 × 2)NEEDS D ≠ 0

\begin{gathered} D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}, \quad D_x = \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix}, \quad D_y = \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix} \\ x = \frac{D_x}{D}, \quad y = \frac{D_y}{D} \end{gathered}

1. Standard FormWrite each equation as

ax + by = c

, with the unknowns lined up.

2. One, None or ManyD ≠ 0: one solution. D = 0: none or infinitely many (check the ranks).

3. Solve and CheckFind the values, then put them back into every equation.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Solve $2x + 3y = 13$ and $x - y = -1$ by Cramer's rule.

STEP 01 • THE DETERMINANT D

D = \begin{vmatrix} 2 & 3 \\ 1 & -1 \end{vmatrix} = (2)(-1) - (3)(1) = -5 \neq 0

D is not zero, so there is exactly one solution.

STEP 02 • REPLACE EACH COLUMN WITH THE CONSTANTS

D_x = \begin{vmatrix} 13 & 3 \\ -1 & -1 \end{vmatrix} = -13 + 3 = -10, \qquad D_y = \begin{vmatrix} 2 & 13 \\ 1 & -1 \end{vmatrix} = -2 - 13 = -15

STEP 03 • DIVIDE AND CHECK

x = \frac{-10}{-5} = 2, \qquad y = \frac{-15}{-5} = 3

2(2) + 3(3) = 13 \ \checkmark \qquad 2 - 3 = -1 \ \checkmark

FINAL CONCLUSION

x = 2, y = 3

(x, y) = (2, 3)

The solver below opens on this pair and draws the two lines meeting at (2, 3).

LABORATORY

Interactive Equation Solver

LABORATORY

Simultaneous Equations Engine

Number of equations

Method

Type 0 for a missing term and a negative number for a minus sign (x − y = −1 is 1, −1, −1).

Common Questions

What is Cramer's rule?

For a system with a non-zero determinant D, each unknown is a ratio of determinants:

x = \frac{D_x}{D}

, where

D_x

is D with the x-column replaced by the constants. The same works for y and z.

When does a system of equations have no solution?

When the equations contradict each other: for two equations, the lines are parallel. Then D = 0 and the rank of the coefficient matrix is less than the rank of the augmented matrix.

When are there infinitely many solutions?

When one equation is just a multiple (or combination) of the others, so the lines or planes coincide. D = 0 and both ranks are equal but less than the number of unknowns.

Which method should I use: substitution, elimination or Cramer?

Substitution is quick when one equation already gives x or y easily. Elimination works for any size and is the basis of Gauss elimination. Cramer's rule is neat for 2 or 3 equations but needs D ≠ 0.

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UNIT 7 • ALGEBRA

System of Equations
Cramer, Elimination, Substitution

CRAMER'S RULE (2 × 2)NEEDS D ≠ 0

\begin{gathered} D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}, \quad D_x = \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix}, \quad D_y = \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix} \\ x = \frac{D_x}{D}, \quad y = \frac{D_y}{D} \end{gathered}

1. Standard FormWrite each equation as

ax + by = c

, with the unknowns lined up.

2. One, None or ManyD ≠ 0: one solution. D = 0: none or infinitely many (check the ranks).

3. Solve and CheckFind the values, then put them back into every equation.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Solve $2x + 3y = 13$ and $x - y = -1$ by Cramer's rule.

STEP 01 • THE DETERMINANT D

D = \begin{vmatrix} 2 & 3 \\ 1 & -1 \end{vmatrix} = (2)(-1) - (3)(1) = -5 \neq 0

D is not zero, so there is exactly one solution.

STEP 02 • REPLACE EACH COLUMN WITH THE CONSTANTS

D_x = \begin{vmatrix} 13 & 3 \\ -1 & -1 \end{vmatrix} = -13 + 3 = -10, \qquad D_y = \begin{vmatrix} 2 & 13 \\ 1 & -1 \end{vmatrix} = -2 - 13 = -15

STEP 03 • DIVIDE AND CHECK

x = \frac{-10}{-5} = 2, \qquad y = \frac{-15}{-5} = 3

2(2) + 3(3) = 13 \ \checkmark \qquad 2 - 3 = -1 \ \checkmark

FINAL CONCLUSION

x = 2, y = 3

(x, y) = (2, 3)

The solver below opens on this pair and draws the two lines meeting at (2, 3).

LABORATORY

Interactive Equation Solver

LABORATORY

Simultaneous Equations Engine

Number of equations

Method

Type 0 for a missing term and a negative number for a minus sign (x − y = −1 is 1, −1, −1).

Common Questions

What is Cramer's rule?

For a system with a non-zero determinant D, each unknown is a ratio of determinants:

x = \frac{D_x}{D}

, where

D_x

is D with the x-column replaced by the constants. The same works for y and z.

When does a system of equations have no solution?

When the equations contradict each other: for two equations, the lines are parallel. Then D = 0 and the rank of the coefficient matrix is less than the rank of the augmented matrix.

When are there infinitely many solutions?

When one equation is just a multiple (or combination) of the others, so the lines or planes coincide. D = 0 and both ranks are equal but less than the number of unknowns.

Which method should I use: substitution, elimination or Cramer?

PreviousQuadratic Equation

Unit 7•48 of 63

NextAP & GP

System of Equations Cramer, Elimination, Substitution

Step-by-Step Worked Example

x = 2, y = 3

Interactive Equation Solver

Simultaneous Equations Engine

Common Questions

NETZ Engine

System of Equations Cramer, Elimination, Substitution

Step-by-Step Worked Example

x = 2, y = 3

Interactive Equation Solver

Simultaneous Equations Engine

Common Questions

System of Equations
Cramer, Elimination, Substitution

System of Equations
Cramer, Elimination, Substitution