Examples with solutions for Linear Function y=mx+b: Understanding the concept of slope

Exercise #1

Which best describes the function below?

y=23x y=2-3x

Video Solution

Step-by-Step Solution

To determine the characteristic of the function y=23x y = 2 - 3x , we will evaluate the slope:

  • The given function is in the form y=mx+b y = mx + b , which indicates a linear equation. Here, y=23x y = 2 - 3x can be rearranged as y=3x+2 y = -3x + 2 , showing that m=3 m = -3 .
  • The slope m m is 3-3.
  • In a linear function, the sign of the slope m m determines the function's behavior:
    • If the slope m m is positive (m>0 m > 0 ), the function is increasing.
    • If the slope m m is negative (m<0 m < 0 ), the function is decreasing.
    • If the slope m=0 m = 0 , the function is constant.
  • Since m=3 m = -3 , which is negative, we conclude that the function is decreasing.

Therefore, the function described by y=23x y = 2 - 3x is decreasing.

Answer

The function is decreasing.

Exercise #2

A linear function with a slope of 5 passes through the point (2,4) (2,4) .

Choose the equation that represents this function.

Video Solution

Step-by-Step Solution

To solve for the equation of a line given a slope and a point:

  • Step 1: Use the point-slope form of the linear equation: yy1=m(xx1) y - y_1 = m(x - x_1) .
  • Step 2: Substitute the given slope m=5 m = 5 and point (2,4) (2, 4) into the equation.
  • Step 3: Simplify and rearrange to convert into slope-intercept form y=mx+b y = mx + b .

Substituting into the point-slope form, we have:

y4=5(x2) y - 4 = 5(x - 2)

Distribute the 5 across the terms in the parentheses:

y4=5x10 y - 4 = 5x - 10

Add 4 to both sides to solve for y y :

y=5x10+4 y = 5x - 10 + 4

This simplifies to:

y=5x6 y = 5x - 6

Therefore, the equation of the line is y=5x6 y = 5x - 6 .

The correct choice among the options given is y=5x6 y = 5x - 6 .

Answer

y=5x6 y=5x-6

Exercise #3

A linear function with a slope of 6 passes through the point (1,4) (1,-4) .

Which equation represents the function?

Video Solution

Step-by-Step Solution

To solve the problem, we will determine the equation of a line using the point-slope form. The general formula for a line in point-slope form is:

yy1=m(xx1) y - y_1 = m(x - x_1)

Given the slope m=6 m = 6 and the point (1,4) (1, -4) , we can substitute these values into the formula:

y(4)=6(x1) y - (-4) = 6(x - 1)

Simplifying the equation, we get:

y+4=6x6 y + 4 = 6x - 6

Now, we want to express this in slope-intercept form y=mx+b y = mx + b . So, we solve for y y :

y=6x64 y = 6x - 6 - 4

Finally, combining like terms gives us the equation:

y=6x10 y = 6x - 10

Therefore, the equation that represents the function is y=6x10 y = 6x - 10 .

The correct answer choice is:

: y=6x10 y=6x-10

Answer

y=6x10 y=6x-10

Exercise #4

A linear function has a slope of 1 and passes through the point (6,13) (6,13) .

Choose the equation that represents this function.

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Use the point-slope form with the given point and slope.

  • Step 2: Simplify the equation to slope-intercept form.

  • Step 3: Identify the correct equation from the options.

Now, let's work through each step:

Step 1: Apply the point-slope form formula:
Given the point (6,13) (6, 13) and the slope m=1 m = 1 , the point-slope form is:
y13=1(x6) y - 13 = 1(x - 6)

Step 2: Simplify to get the equation in slope-intercept form:
y13=1(x6)y13=x6y=x6+13y=x+7 y - 13 = 1 \cdot (x - 6) \\ y - 13 = x - 6 \\ y = x - 6 + 13 \\ y = x + 7

Step 3: Compare to find the correct answer:
From the simplified equation y=x+7 y = x + 7 , the correct choice is:

y=x+7 y=x+7

Therefore, the equation representing the function is y=x+7 y = x + 7 .

Answer

y=x+7 y=x+7

Exercise #5

A linear function with a slope of 6 passes through the point (1,1) (1,1) .

Which equation represents the function?

Video Solution

Step-by-Step Solution

To solve the problem of finding the equation of the linear function, we will use the point-slope form, which is:

yy1=m(xx1) y - y_1 = m(x - x_1)

Step-by-step:

  • Step 1: Identify given information: The slope m=6 m = 6 and the point (x1,y1)=(1,1) (x_1, y_1) = (1, 1) .

  • Step 2: Substitute the slope and point into the point-slope form:

    y1=6(x1) y - 1 = 6(x - 1)
  • Step 3: Simplify the equation:

    y1=6x6 y - 1 = 6x - 6
  • Step 4: Solve for y y to express in slope-intercept form y=mx+b y = mx + b :

    y=6x6+1 y = 6x - 6 + 1
  • Step 5: Simplify the right-hand side:

    y=6x5 y = 6x - 5

Thus, the equation of the linear function is y=6x5 y = 6x - 5 .

Answer

y=6x5 y=6x-5

Exercise #6

A linear function has a slope of -3 and passes through the point (6,3) (-6,-3) .

Choose the equation that represents the function.

Video Solution

Step-by-Step Solution

To determine the equation of the given linear function, follow these steps:

  • Step 1: Identify the key information: the slope m=3 m = -3 and the point (6,3) (-6, -3) .
  • Step 2: Use the point-slope form, yy1=m(xx1) y - y_1 = m(x - x_1) .
  • Step 3: Substitute the values m=3 m = -3 , x1=6 x_1 = -6 , and y1=3 y_1 = -3 into the formula.
  • Step 4: Solve for y y to convert to slope-intercept form y=mx+b y = mx + b .

Now, let's go through the process:

Use the point-slope form:

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

Simplify the equation:

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

Distribute the slope on the right side:

y+3=3x18 y + 3 = -3x - 18

Subtract 3 from both sides to solve for y y :

y=3x183 y = -3x - 18 - 3

which simplifies to:

y=3x21 y = -3x - 21

This equation, y=3x21 y = -3x - 21 , matches the first choice in the provided options.

Therefore, the equation that represents the function is y=3x21 y = -3x - 21 .

Answer

y=3x21 y=-3x-21

Exercise #7

What is the slope of a straight line that passed through the points (0,4),(5,6) (0,4),(-5,6) ?

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the given information about the points.
  • Step 2: Use the slope formula to find the slope.
  • Step 3: Calculate and simplify.

Now, let's compute the slope:

Step 1: The points given are (0,4)(0, 4) and (5,6)(-5, 6).

Step 2: Apply the slope formula:

The slope m m is given by:

m=y2y1x2x1 m = \frac{y_2 - y_1}{x_2 - x_1}

Substitute the known values:

y2=6,  y1=4,  x2=5,  x1=0 y_2 = 6, \; y_1 = 4, \; x_2 = -5, \; x_1 = 0 m=6450=25 m = \frac{6 - 4}{-5 - 0} = \frac{2}{-5}

Step 3: Simplify the expression:

m=25 m = -\frac{2}{5}

Thus, the slope of the line passing through the points (0,4)(0, 4) and (5,6)(-5, 6) is 25-\frac{2}{5}.

Therefore, the solution to the problem is 25 -\frac{2}{5} .

Answer

25 -\frac{2}{5}

Exercise #8

Calculate the slope of the line that passes through the points (4,1),(2,5) (4,1),(2,5) .

Video Solution

Step-by-Step Solution

Remember the formula for calculating a slope using points:

Now, replace the data in the formula with our own:

(51)(24)=42=2 \frac{(5-1)}{(2-4)}=\frac{4}{-2}=-2

Answer

-2

Exercise #9

Calculate the slope of a straight line that passes through the points (6,1),(2,4) (-6,1),(2,4) .

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the coordinates of the given points.

  • Step 2: Substitute these values into the slope formula.

  • Step 3: Simplify to find the slope.

Now, let's work through each step:
Step 1: Given points are (6,1) (-6, 1) and (2,4) (2, 4) . Thus, we have:
x1=6,y1=1 x_1 = -6, y_1 = 1 , x2=2,y2=4 x_2 = 2, y_2 = 4 .
Step 2: Apply the slope formula:
m=y2y1x2x1=412(6) m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 1}{2 - (-6)}
Step 3: Simplify:
Calculate the numerator: 41=34 - 1 = 3.
Calculate the denominator: 2(6)=2+6=82 - (-6) = 2 + 6 = 8.
Thus, the slope mm is:
m=38 m = \frac{3}{8}

Therefore, the solution to the problem is 38 \frac{3}{8} .

Answer

38 \frac{3}{8}

Exercise #10

What is the slope of a straight line that passes through the points (0,0),(8,2) (0,0),(-8,2) ?

Video Solution

Step-by-Step Solution

To solve the problem, remember the formula to find the slope using two points

 

Now, we replace the given points in the calculation:

 (02)(0(8)=28=14 \frac{(0-2)}{(0-(-8)}=\frac{-2}{8}=-\frac{1}{4}

Answer

14 -\frac{1}{4}