Hints for solving the problems (5.1)

Solving by Graphing

Problems 1-5

Look at the problem and name the two lines that cross at the ordered pair given in the problem. This shows that the two equations have one solution.

It also means that the order pair can be substituted into both equations to give a true statement.

When a system has no solution, you pick the two lines that are parallel.

There is also another type of system. This system has two lines that coincide (one equation is the multiple of the other). These two equations have an infinite number of solutions. Which means all ordered pairs that work for one, works also for the other?

Intercepting lines are two equations that have one solution. They have different slopes and the same or different y-intercepts.

Parallel lines are two equations that have no solution; they have the same slopes, but different y-intercepts.

Coinciding lines are two equations that have an infinite number of solutions. They have the same slopes and the same y-intercepts.

Problem 5-22

Problem No. 7 – Graph each equation to find the solution of this system.

          2x + y = 4

          3x - 4y = 6

Graph each equation by using the slope-intercept form. (y = mx + b)

           2x + y = 4          3x - 4y = 6

           y = -2x + 4        -4y = -3x + 6

                                     y = (3/4)x - 6/4

            m = -2/1                 m = ¾

            b = 4 (0, 4)            b = -3/2 (0, -1 ½)

Now plot each equation on graph paper and draw the straight line between at least two points. You will see that the two equations cross at the point (2, 0).

So the answer is the intersecting point at (2, 0).

Problems 23-28

To draw a line on the graph which shows no solution, you need to draw a line parallel to the given line.

To draw a line on the graph which shows infinite number of solutions, you will draw a line right on top of the line given. One way to show this is to draw a dotted line just above or below it. This shows that there are two lines going through the same points.

To draw a line that goes through a selected point on the graph, put a dot on that point and draw any other line through that point. Make sure you don’t just trace over the same line given. (Must have different slopes)

Solution by Algebra

Problems 29-42

Problem No. 32 – Use the substitution method to solve this system.

        4x - 3y = -7

         x + 2y = 12

To solve these problems, using substitution, take one of the equations and solve one unknown in terms of the other. It is easier if you use the equation that has a numerical coefficient of one. Then substitute that into the other equation, not the one you just solved for the variable. When get an answer for one of the variables, substitute it into one of the original equations. The answer is an ordered pair that should be check, using both equations.

         x + 2y = 12       x = -2y + 12      Solving for x

        4(-2y + 12) - 3y  =  -7        Substituting x in the first equation.

Now you have an equation with just one variable (y).

            -8y + 48 - 3y = -7       -11y + 48 = -7

                                                -11y = -55      y = 5

The quick way is to substitute the 5 into the equation that you solved for x at the beginning.

                        x = -2(5) + 12      x = -10 + 12      x = 2

        The answer is the ordered pair: (2, 5)

Make sure you check the answer in both of the original equations. The answer must produce an equal statement in both original equations.

Such as:       4(2) - 3(5) = -7       8 - 15 = -7     True

                         2 + 2(5) = 12      2 + 10 = 12    True

Problems 43-56

Problem No. 43 – Use the elimination method to solve this system.

                   x - y = 3

                   x + y = 5

To solve problems, using the elimination method, add or subtract the equations to eliminate one of the variables. This will now have only one variable left to solve. After solving it, substitute the answer into either of the original equations to get the other answer.

                   x  - y = 3

                   x + y = 5      Add the two equations to eliminate the y.

                  2x      = 8

                     x     = 4      Substitute the four for x in the second equation.

               4 + y = 5      y = 5 - 4      y = 1

     The answer is the order pair: (4, 1)

Be sure to check the (4, 1) in both original equations to see if it produces a true statement. If it doesn’t then the order pair (4, 1) is wrong. Start again with the elimination process.

Problem No. 51 – Use the elimination method to solve this system.

                    5x - 8y = 10

                    3x + 4y = 6

Sometimes you can’t just add to eliminate a variable. Then you have to determine the LCM (Lowest Common Multiple) for each variable.

    The LCM for x is: 15

The LCM for y is: 32 Usually you would use the smallest LCM, but in this case, use the LCM of 32 because the y’s have opposite signs.

Multiply both sides of each equation by a number, not necessarily the same, to get two opposite numerical coefficients.

               4(5x - 8y = 10)        20x  - 32y = 40

                8(3x + 4y = 6)        24x + 32y = 48

                                               44x           = 88

                                                        x    =    2

Substitute the two for x into either equation.

                 5(2) - 8y = 10         10 - 8y = 10      -8y = 0      y = 0

    The answer is the ordered pair: (2, 0)

Substitute the ordered pair (2,0) into both equations to see if it produces a true statement. If it does the answer is (2, 0). If it doesn’t, start over.