A system of equations is a set of two or more equations that you deal with together. Solving a system of equations means finding the values that satisfy all the equations in the system simultaneously. There are various methods to solve systems of equations, including graphing, substitution, and elimination.
Methods of Solving Systems of Equations
- Graphing: Plot each equation on a graph and identify where the lines intersect. The intersection points are the solutions.
- Substitution: Solve one equation for one variable and then substitute that expression into the other equation.
- Elimination: Add or subtract the equations to eliminate one variable, making it easier to solve for the other.
Example 1: Solving by Substitution
Solve the system of equations: y = 2x + 3 and y = x – 1.
Substitute y in the second equation with 2x + 3 from the first equation: 2x + 3 = x – 1.
Solve for x: x = -4. Then substitute x back into one of the equations to find y: y = -4 – 1, so y = -5.
The solution is x = -4, y = -5.
Example 2: Solving by Elimination
Solve the system of equations: 2x + 3y = 5 and 4x – y = 11.
First, multiply the second equation by 3: 12x – 3y = 33.
Add this equation to the first equation: 2x + 3y = 5 and 12x – 3y = 33, getting 14x = 38.
Solve for x: x = 38 / 14. Then substitute x back into one of the equations to find y.
The solution is x = 38 / 14 and y can be found by substituting this value into one of the original equations.
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