SOLVE SYSTEMS OF LINEAR EQUATIONS BY GRAPHING

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What is solution in systems of linear equations ?

The point of intersection is known as solution. In general , two lines may meet at a point, they meet at infinitely many points, they will never meet.

If two lines are intersecting at one point, there is unique solution.
If two lines are parallel, they will never meet. So no solution.
If two lines are coincident, then they will touch each other at infinite number of points. So, infinitely many solutions.

To draw the graph of the line, we have different ways. Here we follow of finding intercepts and drawing the graph.

Find the simultaneous solution of the following pairs of equations using graphical methods:

Problem 1 :

y = x + 1

y = 2x - 3

Solution :

y = x + 1

x-intercept :

Put y = 0

0 = x + 1

x = -1

y -intercept :

Put x = 0

y = 0 + 1

y = 1

y = 2x - 3

x-intercept :

Put y = 0

0 = 2x - 3

2x = 3

x = 3/2

(3/2, 0)

y -intercept :

Put x = 0

y = 2(0) - 3

y = -3

(0, -3)

The lines are intersecting at (4, 5). So, the solution is (4, 5).

Problem 2 :

y = x + 4

y = -x + 2

Solution :

y = x + 4

x-intercept :

Put y = 0

0 = x + 4

x = -4

(-4, 0)

y-intercept :

Put x = 0

y = 0 + 4

y = 4

(0, 4)

By joining the points (-4, 0) and (0, 4) we can get the first line.

y = -x + 2

x-intercept :

Put y = 0

0 = -x + 2

-x = -2

x = 2

(2, 0)

y-intercept :

Put x = 0

y = -x + 2

y = -0 + 2

y = 2

(0, 2)

By joining the points (2, 0) and (0, 2), we get the second line.

The point of intersection is (-1, 3). So, the solution is (-1, 3).

Problem 3 :

y = x + 2

y = -2x + 5

Solution :

y = x + 2

x-intercept :

Put y = 0

0 = x + 2

x = -2

(-2, 0)

y-intercept :

Put x = 0

y = 0 + 2

y = 2

(0, 2)

By joining the points (-2, 0) and (0, 2) we can get the first line.

y = -2x + 5

x-intercept :

Put y = 0

0 = -2x + 5

-2x = -5

(5/2, 0)

y-intercept :

Put x = 0

y = -2(0) + 5

y = 5

(0, 5)

By joining the points (-2, 0) and (0, 2) we can get the second line.

So, the solution is (1, 3).

Problem 4 :

y = 2x - 2

y = 1 - x

Solution :

y = 2x - 2

x-intercept :

Put y = 0

0 = 2x - 2

2x = 2

x = 1

(1, 0)

y-intercept :

Put x = 0

y = 2(0) - 2

y = -2

(0, -2)

By joining the points (1, 0) and (0, -2), we draw the first line.

y = 1 - x

x-intercept :

Put y = 0

0 = 1 - x

x = 1

(1, 0)

y-intercept :

Put x = 0

y = 1 - 0

y = 1

(0, 1)

By joining the points (1, 0) and (0, 1), we draw the second line.

So, the solution is (1, 0).

Problem 5 :

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

solving-system-of-equations-by-graphing-q1

Solution :

x - 3y = 6 -----(1)

2x - 3y = 3 ----(2)

From (1), x = 6 + 3y

Applying the value of x in (2), we get

2(6 + 3y) - 3y = 3

12 + 6y - 3y = 3

12 + 3y = 3

3y = 3 - 12

3y = -9

y = -9/3

y = -3

Applying the value of y in (1), we get

x = 6 + 3(-3)

x = 6 - 9

x = -3

The solution should be (-3, -3). But observing the graph it says (3, -1) is the solution and that is the error.

Problem 6 :

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

Solution :

y = 2x - 1 -----(1)

y = x + 1----(2)

(1) = (2)

2x - 1 = x + 1

2x - x = 1 + 1

x = 2

Applying the value of x, we get

y = 2 + 1

y = 3

So, the solution should be (2, 3).

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SOLVE SYSTEMS OF LINEAR EQUATIONS BY GRAPHING

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