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Sever21 [200]
1 year ago
5

← How many solutions does this linear system have y = x+2 6x - 4y = -10

Mathematics
1 answer:
defon1 year ago
8 0

Answer:

This linear system has one solution.

Step-by-step explanation:

First equation: y = x + 2

Second equation: 6x - 4y = -10

Let's change the second equation in slope-intercept form y = mx + b.

<u>Slope-intercept form</u>

y = mx + b

m ... slope

b ... y-intercept

6x - 4y = -10

6x + 10 = 4y

\frac{6}{4}x + \frac{10}{4} = y

\frac{3}{2}x + \frac{5}{2} = y

If two lines have the <em>same slope </em>but <em>different y-intercept</em>, they are parallel - <u>system has no solutions</u>.

If two lines have the <em>same slope</em> and the <em>same y-intercept</em>, they are the same line and are intersecting in infinite many points - <u>system has infinite many solutions</u>.

If two lines have <em>different slopes</em> then they intersect in one point - <u>system has one solution</u>.

We see that lines have different slopes. First line has slope 1 and the other line has slope \frac{3}{2}. So the system has one solution.

You can also check this by solving the system.

Substitute y in second equation with y from first.

6x - 4y = -10

6x - 4(x + 2) = -10

Solve for x.

6x - 4x - 8 = -10

2x = -2

x = -1

y = x + 2

y = -1 + 2

y = 1

The lines intersect in point (-1, 1). <-- one solution

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For the linear function f(x) = -2x+1 , find the following
Leya [2.2K]

Given:

The function is:

f(x)=-2x+1

To find:

a. Slope and Y-intercept.

b. Is the function increasing, decreasing, or constant, justify your answer.

c. Graph the function, label x, and y-intercepts.

Solution:

a. We have,

f(x)=-2x+1                  ...(i)

The slope intercept form of a line is:

y=mx+b                    ...(ii)

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b. If the slope of a linear function is negative, then the function is decreasing.

If the slope of a linear function is positive, then the function is increasing.

If the slope of a linear function is 0, then the function is constant.

The slope of the linear function is negative.

Therefore, the function is decreasing.

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f(x)=-2x+1

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x=0.5

So, the x-intercept is at \left(0.5,0\right).

Plot the points (0,1), \left(0.5,0\right) and connect them by a straight line as shown below:

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