Solving Pairs of Linear Equations—Lots of Ways! Section 5.2 Solving Pairs of Linear Equations—Lots of Ways!

SOVLING SIMULATNEOUS LINEAR EQUATIONS Consider the pair (or system) of linear equations of the form where x and y are the unknowns and the coefficients a, b, p in the first equation and c, d, q in the second equation are given constants. A solution of this system is simply a pair (x, y) of numbers that make both equations true at the same time (simultaneously).

POSSIBILITIES FOR THE SOLUTION TO A LINEAR PAIR OF EQUATIONS There are just three possibilities for the solution to a linear pair of equations: Exactly one solution. Graphically, this is two intersecting lines. No solution. Graphically, this is two parallel lines. Infinitely many solutions. Graphically, this is two coincident lines (two lines—one on top of the other).

GRAPHICAL METHOD FOR SOLVING Solve both equations for y; that is, make each equation look like y = “something.” Graph each equation (on your calculator). Use the “intersection” feature of your calculator to find the x and y that solve the system.

ALGEBRAIC SOLUTION— METHOD OF ELIMINATION Add an appropriate constant multiple of the first equation to the second equation. (The idea is to choose the constant multiple that serves to eliminate the variable x from the second equation.) Solve the resulting equation for y. Substitute the value for y back into one of the original equations to find x.

SYMBOLIC SOLUTIONS Given the system of equations The solutions are:

DETERMINANTS The fractions on the previous slide both have denominators of ad − bc. This is the value of the 2-by-2 (or 2 × 2) determinant defined by NOTE: Δ is upper-case Greek letter delta.

DETERMINANT SOLUTION TO A PAIR OF LINEAR EQUATIONS Given the system of equations The solutions are given by

REMARKS ON DETERMINANT SOLUTIONS Consider The value for Δ gives information about the solutions to the system If Δ ≠ 0, the system has exactly one solution. If Δ = 0, the system either has no solutions or has infinitely many solutions.

MATRIX Definition: A matrix is an array (table) of numbers. EXAMPLES:

MATRICES AND SYSTEMS OF EQUATIONS The solution to the system of equations is determined by its coefficient matrix A and its constant matrix B.

MATRICES AND SYSTMES (CONTINUED) We can abbreviate the system by writing or simply

SOLVING A SYSTEM OF LINEAR EQUATIONS USING MATRICES The solution to the system of equations is given by the matrix equation The 2 × 1 matrix is called the solution matrix.