3.2 Connections to Algebra Solving systems of linear equations AND Equations of lines

Solutions to linear systems One:if lines intersect at a point (different m) None:if lines are parallel (same m, different b) Many:if lines are conincident (same m, same b)

If two distinct lines intersect, then their intersection is exactly one point. Graphing Substitution Combination/Elimination You can find the point of intersection by:

Graphing Can easily graph y = mx +b Can easily graph Ax + By = C Good for approximating Not very accurate Not always given “nice” equations Time consuming to create

Graphing x + y = 4 Graph using the intercepts Graph using the slope and y-intercept

Substitution: Gives exact answer Uses common algebraic techniques Does not work well with coefficients that are not equal to one Not easy with fractions Look for one of the variables to have a coefficient of 1 (no number)

Substitution Solve one of the equations for either x or y and substitute into the other equation. 5x + 2y = -1 x + 3y = 5 Rewrite solving for x: x = 5 – 3y Now substitute the expression in for x into the other equation 5(5 – 3y) + 2y = -1 Solve for remaining variable y = 2 Substitute into the rewritten equation And solve for the remaining variable. X = -1

Combination/Elimination Gives exact answer Easier to use when fractions are involved Uses common algebraic techniques Must find common multiples Easy to make careless mistakes with signs and not multiplying to everything on a side

Combination and Elimination 5x + 2y = -1 -2x + 3y = 5 Must get coefficients of either x or y to be equal and opposite in value. This is done by multiplying both sides of each equation by a common multiple. Once one of the coefficients is equal and opposite, ADD both Equations together and the variable will “drop out”. Solve for the remaining variable.

Combination/Elimination 5x + 2y = -1 -2x + 3y = 5 2(5x + 2y) = (-1)2 5(-2x + 3y) = (5)5 10x + 4y = x + 15y = 25 19y = 23

Combination/Elimination 5x + 2y = -1 -2x + 3y = 5 Now repeat the process to eliminate the other variable.

Combination/Elimination with FRACTIONS You can easily do the same technique with linear equations that involve fractions. GET RID OF THE FRACTIONS by multiplying through the entire equation by a Least Common Multiple. You should now have no fractions, thus proceed to solve.

Combination/Elimination with Fractions

Finding the equation of a line y = mx + bslope intercept Ax + By = Cstandard form y - y 1 = m(x – x 1 )point-slope form x = avertical y = bhorizontal

Finding equations of lines parallel to a given line Remember slopes of parallel lines are equal, so steal the slope and use the given point to write the equation.

Finding equations of lines perpendicular to a given line Remember slopes of perpendicular lines are opposite reciprocals, so steal the slope, take its opposite reciprocal as the new slope and use the given point to write the equation.

Parallel and Perpendicular Postulates PARALLEL POSTULATE: If there is a line and a point not on the line, then there is exactly ONE LINE through the point parallel to the given line. PERPENDICULAR POSTULATE: If there is a line and a point not on the line, then there is exactly ONE LINE through the point perpendicular to the given line.