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Find the point of intersection of the following pair of equations, then sketch your solutions indicating all point where the equations intersect both sets of axes: a) y = 2x – 1 and x + 2y = 5b) ^2 + ^2 = 4 and 3x + y = 2

Sagot :

Answer:

a) Intersection of the equations (1.4, 1.8)

b) Intersection of the equations (0, 2) and (1.2, -1.6)

Explanation:

Part a) y = 2x – 1 and x + 2y = 5

To find the intersection point, let's replace the first equation on the second one, so

x + 2y = 5

x + 2(2x - 1) = 5

Now, we can solve the equation for x

x + 2(2x) - 2(1) = 5

x + 4x - 2 = 5

5x - 2 = 5

5x - 2 + 2 = 5 + 2

5x = 7

5x/5 = 7/5

x = 1.4

Then, replace x = 7/5 on the first equation

y = 2x - 1

y = 2(1.4) - 1

y = 2.8 - 1

y = 1.8

Then, the graph of the lines is:

Where the intersection points with the axes for y = 2x - 1 are (0, -1) and (0.5, 0) and the intersection points with the axes of x + 2y = 5 are (0, 2.5) and (5, 0)

Part b) ^2 + ^2 = 4 and 3x + y = 2

First, let's solve 3x + y = 2 for y, so

3x + y - 3x = 2 - 3x

y = 2 - 3x

Then, replace y = 2 - 3x on the first equation and solve for x

x² + y² = 4

x² + (2 - 3x)² = 4

x² + 2² - 2(2)(3x) + (3x)² = 4

x² + 4 - 12x + 9x² = 4

10x² - 12x + 4 = 4

10x² - 12x + 4 - 4 = 0

10x² - 12x = 0

x(10x - 12) = 0

So, the solutions are

x = 0

or

10x - 12 = 0

10x = 12

x = 12/10

x = 1.2

Replacing the values of x, we get that y is equal to

For x = 0

y = 2 - 3x

y = 2 - 3(0)

y = 2 - 0

y = 2

For x = 1.2

y = 2 - 3(1.2)

y = 2 - 3.6

y = -1.6

Therefore, the intersection points are (0, 2) and (1.2, -1.6)

Then, the graph of the functions are:

Since ^2 + ^2 = 4 is a circle with radius 2, the intersection points with the axes are (2,0), (0, -2), (-2, 0) and (0, 2). Additionally, the intersections potins with the axis of the line 3x + y are (0, 2) and (0.667, 0)

View image AnapaulaR391980
View image AnapaulaR391980
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