Linear Equations and Systems: Linear System in Two Unknown
Solve system of linear equation
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A linear system in two unknowns refers to a system of two equations,
The general form of such a system is:
Here, x and y are the variables, and a1,b1,c1,a2,b2,c2 are constants. The goal is to find the values of x and y that satisfy both equations simultaneously.
3 ways to solve systems of linear equations: substitution, elimination, and graphing.
But lets focus on Substitution and Elimination;
Substitution Method
Solve one of the equations for one variable and substitute it into the other equation. This method can be particularly useful when one of the equations is already solved for a variable.
For Example :
Find the solution to the system equation
y = x + 4
2x — 3y = 7
Since x is already solved in the first equation, lets substituting x+ 4 for y
2x + 3y = 7
2 — 3(x + 4) = 7
Start distributing the 3.
2x — 3x -12 = 7
Combine like terms
-x — 12 = 7
Add 12 both sides
-x — 12 + 12 = 7 + 12
-x = 19
Multiply both sides by −1.
x = — 19
To find y, we’ll plug in −19 for x in the first equation.
y = x + 4
y = — 19 + 4
y = — 15
The unique solution is (−19, − 15).
Elimination Method (or Addition Method)
Multiply one or both equations by a constant to make the coefficients of one variable the same (but with opposite signs), then add or subtract the equations to eliminate one variable.
For Example :
Find the solution to the system equation
- 3x + y = — 7 (Equation 1)
x + 2y = 7(Equation 2)
Multiply the result above by 2 so that the y-terms will cancel when
we subtract the equations.
2(−3x + y) = 2(−7)
-6x + 2y = — 14
Now we’ll subtract the equations.
-6x + 2y = — 14
x + 2y = 7
Combine like terms.
-6x — x + 2y — 2y = — 14–7
-7x = -21
Divide both sides by −7.
x = 3
To solve, we’ll plug in 3 for x in the original first equation.
- 3x + y = — 7
- 3(3) + y = — 7
y = -7 + 9
y = 2
The solution is (3,2).
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