Simplify the following algebraic expressions.
- 6x + 5 + 12x -6
2(x - 9) + 6(-x + 2) + 4x
3x2 + 12 + 9x - 20 + 6x2 - x
(x + 2)(x + 4) + (x + 5)(-x - 1)
1.2(x - 9) - 2.3(x + 4)
(x2y)(xy2)
(-x2y2)(xy2)
Solution
Group like terms and simplify.
- 6x + 5 + 12x -6 = (- 6x + 12x) + (5 - by
Solution
Use rules of exponential to simplify the numerator first.
(a b2)(a3 b) / (a2 b3) = (a4 b3) / (a2 b3)
Rewrite as follows.
(a4 / a2) (b3 / b3)
Use rule of quotient of exponentials to simplify.
= a2
Rewrite as follows.
(21 x5) / (3 x4) = (21 / 3)(x5 / x4)
Simplify.
= 7 x
(6 x4)(4 y2) / [ (3 x2)(16 y) ]
Multiply terms in numerator and denominator and simplify.
(6 x4)(4 y2) / [ (3 x2)(16 y) ] = (24 x4 y2) / (48 x2 y)
Rewrite as follows.
= (24 / 48)(x4 / x2)(y2 / y)
Simplify.
= (1 / 2) x2 y
Factor 4 out in the numerator.
(4x - 12) / 4 = 4(x - 3) / 4
Simplify.
= x - 3
Factor -5 out in the numerator.
(-5x - 10) / (x + 2) = - 5 (x + 2) / (x + 2)
Simplify.
= - 5
Factor numerator and denominator as follows.
(x2 - 4x - 12) / (x2 - 2x - 24) = [(x - 6)(x + 2)] / [(x - 6)(x + 4)]
Simplify.
= (x + 2) / (x + 4) , for all x not equal to 6
Solve for x the following linear equations.
2x = 6
6x - 8 = 4x + 4
4(x - 2) = 2(x + 3) + 7
0.1 x - 1.6 = 0.2 x + 2.3
- x / 5 = 2
(x - 4) / (- 6) = 3
(-3x + 1) / (x - 2) = -3
x / 5 + (x - 1) / 3 = 1/5
Solution
Divide both sides of the equation by 2 and simplify.
2x / 2 = 6 / 2
x = 3
Add 8 to both sides and group like terms.
6x - 8 + 8 = 4x + 4 + 8
6x = 4x + 12
Add - 4x to both sides and group like terms.
6x - 4x = 4x + 12 - 4x
2x = 12
Divide both sides by 2 and simplify.
x = 6
Expand brackets.
4x - 8 = 2x + 6 + 7
Add 8 to both sides and group like terms.
4x - 8 + 8 = 2x + 6 + 7 + 8
4x = 2x + 21
Add - 2x to both sides and group like terms.
4x - 2x = 2x + 21 - 2x
2x = 21
Divide both sides by 2.
x = 21 / 2
Add 1.6 to both sides and simplify.
0.1 x - 1.6 = 0.2 x + 2.3
0.1 x - 1.6 + 1.6 = 0.2 x + 2.3 + 1.6
0.1 x = 0.2 x + 3.9
Add - 0.2 x to both sides and simplify.
0.1 x - 0.2 x = 0.2 x + 3.9 - 0.2 x
- 0.1 x = 3.9
Divide both sides by - 0.1 and simplify.
x = - 39
Multiply both sides by - 5 and simplify.
- 5(- x / 5) = - 5(2
What is the y intercept of the line - 4 x + 6 y = - 12?
Solution
Set x = 0 in the equation and solve for y.
- 4 (0) + 6 y = - 12
6 y = - 12
y = - 2
y intercept: (0 , - 2)
What is the x intercept of the line - 3 x + y = 3?
Solution
Set y = 0 in the equation and solve for x.
- 3 x + 0 = 3
x = -1
x intercept: (-1 , 0)
What is point of intersection of the lines x - y = 3 and - 5 x - 2 y = - 22?
Solution
A point of intersection of two lines is solution to the equations of both lines. To find the point of intersection of the two lines, we need to solve the system of equations x - y = 3 and -5 x - 2 y = -22 simultaneously. Equation x - y = 3 can be solved for x to give
x = 3 + y
Substitute x by 3 + y in the equation - 5 x - 2 y = -22 and solve for y
-5 (3 + y) - 2 y = - 22
-15 - 5 y - 2 y = - 22
-7 y = - 22 + 15
-7 y = - 7
y = 1
Substitute x by 3 + y in the equation -5 x - 2 y = - 22 and solve for y
x = 3 + y = 3 + 1 = 4
Point of intersection: (4 , 1)
For what value of the constant k does the line - 4 x + k y = 2 pass through the point (2,-3)?
Solution
For the line to pass through the point (2,-3), the ordered pair (2,-3) must be a solution to the equation of the line. We substitute x by 2 abd y by - 3 in the equation.
- 4(2) + k(-3) = 2
Solve the for k to obtain
k = - 10 / 3
What is the slope of the line with equation y - 4 = 10?
Solution
Write the given equation in slope intercept form y = m x + b and identify the slope m.
y = 14
It is a horizontal line and therefore the slope is equal to 0.
What is the slope of the line with equation 2 x = -8?
Solution
The above equation may be written as
x = - 4
It is a vertical line and therefore the slope is undefined.
Find the x and y intercepts of the line with equation x = - 3?
Solution
The above is a vertical line with x intercept only given by
(-3 , 0)
Find the x and y intercepts of the line with equation 3 y - 6 = 3?
Solution
The given equation may be written as
y = 3
The above is a horizontal line with y intercept only given by
(0 , 3)
What is the slope of a line parallel to the x axis?
Solution
A line parallel to the x axis is a horizontal line and its slope is equal to 0.
What is the slope of a line perpendicular to the x axis?
Solution
A line perpendicular to the x axis is a vertical line and its slope is undefined
Step-by-step explanation:
y = -22 and solve for y
-5 (3 + y) - 2 y = - 22
-15 - 5 y - 2 y = - 22
-7 y = - 22 + 15
-7 y = - 7
y = 1
