Question

Explain what it means to solve a system of linear equations.

Short answer

Solving a system of linear equations means finding the set of values for the variables that satisfy all the equations at the same time. This often involves using methods like substitution or elimination to find the values of the variables.

Step-by-step solution

Understanding Systems

A system of linear equations is a set of two or more linear equations that all contain the same set of variables. A solution to the system is an assignment of values to the variables such that all the equations are simultaneously satisfied. In other words, a solution is a point where all the equations in the system intersect when graphed.

Solving Method Explanation - Substitution

One method for solving systems of equations is substitution. This involves solving one of the equations for one of the variables, and then substituting that expression into the other equation. This results in a single equation with just one variable, which can be solved using the usual techniques. The solution can be checked by substituting it back into the original equations to ensure it satisfies them.

Solving Method Explanation - Elimination

Another method for solving systems of equations is elimination. In this method, we multiply one or both of the equations by suitable numbers to make the coefficients on one of the variables the same (or opposites) in both equations. By adding or subtracting the equations, we can eliminate one of the variables, leading to a single equation with a single variable. Once this equation is solved, the resulting value can be substituted back into one of the original equations to find the other variable(s). Once again, the solution can be checked by substiting it back into the original equations.