SOLVING LINEAR EQUATIONS IN TWO VARIABLES GRAPHICALLY WORKSHEET

Solve the following systems of linear equations using graphical method :

Problem 1 :

y = 4x + 3

y = -x - 2

Solution

Problem 2 :

5x + 3y - 9 = 0

x - 3y - 9 = 0

Solution

Problem 3 :

A roofing contractor buys 30 bundles of shingles and 4 rolls of roofing paper for $1040. In a second purchase (at the same prices), the contractor buys 8 bundles of shingles for $256. Find the price per bundle of shingles and the price per roll of roofing paper

Solution

Problem 4 :

Describe and correct the error in solving the system of linear equations.

Solution

On comparing the ratios a₁/a₂, b₁/b₂, c₁/c₂, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident.

Problem 1 :

5 x – 4 y + 8  =  0 and 7 x + 6 y – 9  =  0

Solution

Problem 2 :

9 x + 3 y + 12  =  0 and 18 x + 6 y + 24  =  0

Solution

Problem 3 :

6 x - 3 y + 10  =  0 and 2 x - y + 9  =  0

Solution

Problem 4 :

Consider the system of linear equations

y = ax + 4 and y = bx − 2

where a and b are real numbers. Determine whether each statement is always, sometimes, or never true. Explain your reasoning.

a. The system has infinitely many solutions.

b. The system has no solution.

c. When a < b, the system has one solution.

Solution

Problem 5 :

Compare the slopes and y-intercepts of the graphs of the equations in the linear system

8x + 4y = 12 and 3y = −6x − 15

to determine whether the system has one solution, no solution, or infinitely many solutions. Explain.

Solution

Problem 6 :

You and your friend plant an urban garden. You pay $15.00 for 6 tomato plants and 6 pepper plants. Your friend pays $22.50 for 9 tomato plants and 9 pepper plants. How much does each plant cost?

Solution

Problem 7 :

Find the values of a and b so the system shown has the solution (2, 3). Does the system have any other solutions for these values of a and b? Explain.

12x − 2by = 12

3ax − by = 6

Solution

Problem 8 :

What is the solution of the system of equations?

y = − (2/3)x − 1

4x + 6y = − 6

a)  (-3/2, 0)        b) (0, -1)

c)  no solution        d) infinitely many solution

Solution

On comparing the ratios a₁/a₂, b₁/b₂ and  c₁/c₂, find out whether the following pair of linear equations are consistent or inconsistent.

Example 1 :

3 x + 2 y = 5 and 2 x - 3 y = 7

Solution

Example 2 :

2 x - 3 y = 8 and 4 x - 6 y = 9

Solution

Example 3 :

(3/2) x + (5/3) y = 7 and 9 x - 10 y = 14

Solution

Example 4 :

(4/3)x + 2y = 8 and 2x + 3y = 12

Solution

Example 5 :

5y = 10x + 11

−5y = 5x − 21

Solution

Example 6 :

The pair of linear equations 2x + 3y = 5 and 4x + 6y = 10 is

(a) inconsistent     (b) consistent

(c) dependent consistent       (d) none of these

Solution

Example 7 :

The pair of equations y = 0 and y = –7 has

(a) one solution        (b) two solutions

(c) infinitely many solutions        (d) no solution

Solution

Example 8 :

The pair of equations x = 4 and y = 3 graphically represents lines which are

(a) parallel             (b) intersecting at (3, 4)

(c) coincident        (d) intersecting at (4, 3)

Solution

Example 9 :

A pair of linear equations which has a unique solution x = 2, y = –3 is

(a) x + y = –1; 2x – 3y = –5

(b) 2x + 5y = –11; 4x + 10y = –22

(c) 2x – y = 1 ; 3x + 2y = 0

(d) x – 4y – 14 = 0; 5x – y – 13 = 0

Solution

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