4.1 Solving Systems of Equations in Two Variables
Determine If a Value is a Solution
Replace the x and y values in the equations and see if the values satisfies BOTH equations.
Example: Given
3x + y = 1
y = 1 – 2x
For (2,-5) 3(2) + (-5) | 1 (-5) | 1 – 2(2)
1 = 1 -5 ≠ -3
both do not check
For (0,1) 3(0) + (1) | 1 (1) | 1 – 2(0)
1 = 1 1 =1
both check
Methods to Solve a System of Two Equations and Two Variables
Solve by Graphing
Solve by Substitution
Solve by Elimination
Solve by Graphing
Graph both linear equations
If, they intercept at one point determine the x/y coordinates
If, they don't intersect at one point
Different lines – no solution
Same line – infinite number of solutions
Check your solution
Examples
[20]
x - y = 1
2x + y = 8
[24]
3x + 2y = 8
6x + 4y = 18
[26]
2x + y = -3
8x + 4y = -12
Definitions
Consistent system of equations - system with at least one solution
Dependent system of equations – system with many solutions
Inconsistent system of equations – system with no solutions
Solve by Substitution
1. Isolate (solve for) one of the variables.
2. If both values cancel, if the equation is true (infinite solutions), if false (no solution), stop
3. Substitute that value in the OTHER equation
4. Solve for the unknown
5. Use either original equation and find other value
6. Check your solution
Examples
Solve by Substitution
[44]
x = y +6
x + y = -2
(2,-4)
[54]
2x - y = 6
-2x + 4y = -12
No Solution
Solve by Elimination (aka Addition/Subtraction)
1. Write the equations in standard form ( Ax + By = C )
2. Clear equations of fractions/decimals
3. Determine the LCD of coefficients of x and y, choose one
4. Multiply both equations to make both coefficients be additive inverses for the chosen variable.
5. Add the equations [If both variables are eliminated, check to see if the equation is valid (infinite solutions) or
invalid (no solution)]
6. Solve for the variable left
7. Substitute this value to get the other variable
8. Check your solution
Examples
Solve by elimination
[62]
3x + 2y = 6
5x + 2y = 14
(4,-3)
[78]
4x + 2y = 6
6x = 9 - 3y
{(x.y) | y = -2x + 3}
Hint: If one of the coefficients equals ± 1, it is probably easier to solve by substitution.