Ordered Pairs. Graphing Lines and Linear Inequalities, Solving System of Linear Equations. Cartesian Coordinates System.

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1 Ordered Pairs Graphing Lines and Linear Inequalities, Solving System of Linear Equations Peter Lo All equations in two variables, such as y = mx + c, is satisfied only if we find a value of x and a value of y that make it true. These two values are called Ordered Pairs and denoted as (x, y ). Ma04 Peter Lo 2002 Ma04 Peter Lo Plotting Points Cartesian Coordinates System The Rectangular and Cartesian Coordinates System consists of a horizontal number line, the x-axis, and a vertical line, the y-axis. The intersection of the axes is the Origin. The axes divide the coordinate plane, or the xyplane, into four regions called the Quadrants. Locating a point in the rectangular coordinate system is referred to as plotting or graphing the point. Ma04 Peter Lo Ma04 Peter Lo

2 Intercepts The x-intercept of a line is the point where the line cross the x-axis. The value of x-intercepts is always (x, 0) The y-intercept of a line is the point where the line cross the y-axis. The value of y-intercepts is always (0, y ) Graphing Lines by Plotting Points We know that the solution set to an equation is a line if the equation is linear in two variables that is, there are two variables in the problem, and each variable only appears to a single power, is only in the numerator of fractions, is never under a radical symbol, and is not in the same term as the other variables. To graph lines by plotting points, first make a table of values and then plot each point. Ma04 Peter Lo Ma04 Peter Lo Graphing Lines Using Intercepts Graphing line using intercept is to find the x- and y-intercepts, and since we know that the equation represents a straight line, draw a line between these two points. This method does not always work if the line through the origin (0, 0). Slope The slope of a line is a measure of its steepness. A line with slope whose absolute value is very steep is steep, while a line with a slope whose absolute value is very steep is shallow. The formula for the slope m between two points (x, y ) and (x 2, y 2 ) on a line is: y m = x 2 2 y x Ma04 Peter Lo Ma04 Peter Lo

3 Slope of Parallel and Perpendicular Lines Two lines with slopes m and m 2 are parallel if and only if m = m 2 Two lines with slopes m and m 2 are perpendicular if and only if m m 2 = - Formation of Equation There are four basic methods used to find a equation of a line from a description of it. Two point Form Point-Slop Form Slope-Intercept Form Intercept Form The equation in the form of Ax + By + C = 0 is called a Standard Form. Ma04 Peter Lo Ma04 Peter Lo Comparison of Different Methods Form Information Needed Equation Methods for Graphing a Linear Equation Two Points Form Point-Slope Form Slope-Intercept Form Intercept Form Point: (x, y ) Point: (x 2, y 2 ) Slope: m Point: (x, y ) Slope: m y-intercept: (0, c) x-intercept (b, 0) y-intercept: (0, c) y y y y 2 = x x x x y y = m( x ) 2 y = mx + c x x y + b c = Ma04 Peter Lo 2002 Ma04 Peter Lo

4 Linear Inequality Graphing a Linear Inequality If A, B and C are real numbers with A and B are not zero, then Ax + By C Ax + By C Ax + By < C Ax + By > C is called a Linear Inequality. Ma04 Peter Lo Ma04 Peter Lo Graphing Compound Inequalities. Graph the lines Use a dashed line if the line is not to be included in the solution set (< or >). Use a solid line if the line is to be included in the solution set ( or ). 2. The plane of graph is divided into two or more region by the lines. Find a Test Point for each region. 3. Check to see if the test point satisfy the inequality. 4. Shade the appropriate regions. Graph the solution set to the compound inequality 2x + y 4 and x 2y > 7. Ma04 Peter Lo Ma04 Peter Lo

5 Solving Linear Systems by Graphing When solving a system of equations by graphing, it is very important for the graph to be as accurate as possible. Keep in mind that graph is next exact, and when you solve by graphing, you need to minimize the errors in the graph. After solved a system of equations by graph, MUST check the answer in both of the original equations. Remember that the point of intersection must lie on both lines to satisfy both equations. Solving Linear Systems using Substitution. Solve one of the equations for each variables 2. Substitute this expression into the other equation (the one that did not use in Step ) 3. Solve the resulting equation 4. Substitute this answer back into your expression from Step 5. Check the answer in the original equations. Ma04 Peter Lo Ma04 Peter Lo x + 2y = 6 2x - y = 5 Solving Linear Systems using the Addition Method (Elimination Method). Write the equations in the same form. 2. Multiply the equations by a numbers so that in the resulting equations one of the variables has opposite coefficients for the two equations. 3. Add the equation together term-by-term. 4. Solve the resulting equation. 5. Substitute this answer back into either original equation and solve for the remaining variables. 6. Check your answer in both of the original equations. Ma04 Peter Lo Ma04 Peter Lo

6 3x 2y = 5 7x + 5y = 2 Solving Systems of Linear Equations in Three Variables. Choose two of the three original equations and, using addition or substitution, combine the equations so that one of the three variables is eliminated. 2. Choose another pair of the original equation and, using the same method as in Step, combine these equations to eliminate the same variable as you did in Step. Ma04 Peter Lo Ma04 Peter Lo Solving Systems of Linear Equations in Three Variables 3. Using the results obtained in Step and Step 2, form a new system of two equations in two unknown and solve it using addition or substitution. 4. Substitute the value of the two variables obtained in Step 3 into one of the three original equations and solve for the value of the third variables. 5. Check the answer in all three of the original equation. 3x + 2y 3z = 4 x + 3y 7z = -2 2x y = Ma04 Peter Lo Ma04 Peter Lo

7 Solving Linear Systems using Matrices The augmented matrix and Gaussian elimination can be used to solve a system of n linear equations in n unknowns for an natural number n. To write the augmented matrix, write each equation in standard form, always keeping the variables lined up vertically and constants on the right-hand side of the equation. If a variable is missing, you can either leave the spot for it empty, of include a zero for it. Copy the coefficients into the matrix, remembering the negative signs when appropriate. The position in the matrix tells which variable the number is a coefficient of. Ma04 Peter Lo Steps for Solving Linear Systems using Matrices. Interchanging Row (this corresponds to writing the equations in a different order) 2. Multiplying a row by a non-zero number (this corresponds to multiplying one of the equations by that number) 3. Replacing a row with a sum of it and another row (this corresponds to adding two equations) Ma04 Peter Lo x + y + 2z = 2 x y 4z = 3 3x + 2y 2z = Reference Algebra for College Students (Ch. 3 4) Ma04 Peter Lo Ma04 Peter Lo

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