Example 1: Simplify the expression
2(-4a - 5b) - (8 + b) + b + (-2b + 4) - 5a
Solution to Example1
-
given
2(-4a - 5b) - (8 + b) + b + (-2b + 4) - 5a
-
multiply factors
-8a - 10b - 8 - b + b -2b + 4 - 5a
-
group like terms
- 13a - 12b - 4
Matched Exercise 1 Simplify the expression
2(a - 8b) - (5 - b) + b + (6b - 9) - a
solution
Example 2: Solve the equation
2(-3x - 5) - (8 - x) = -2(2x + 4) + 12
Solution to Example 2
-
given
2(-3x - 5) - (8 - x) = -2(2x + 4) + 12
-
multiply factors
-6x -10 - 8 + x = -4x - 8 + 12
-
group like terms
-5x - 18 = -4x + 4
-
add 18 to both sides
-5x -18 + 18 = -4x + 4 + 18
-
group like terms
-5x = -4x + 22
-
add 4x to both sides
-5x + 4x = -4x + 22 +4x
-
group like terms
-x = 22
-
multiply both sides by -1
x = -22
-
Check the solution
left side:2(-3*(-22) - 5) - (8 - (-22)) = 92
right side:-2(2(-22) +4) + 12 = 92
-
Conclusion
x = -22 is the solution to the given equation
Matched Exercise 2: Solve the equation
2(-x - 5) - (-6 + x) = -3(2x + 4) + 12
solution
/* script-replace-4c7a41ebe650b2c20ebe2a8e80d88f02 */
Example 3: If x > -2, simplify the expression
2| x + 2 | - 3x - (-2 - x) + | 6 - 9 |
Solution to Example 3
-
To simplify the given expression, we need to simplify the terms with absolute value using definition of absolute value.
if x > = 0 , | x | = x
if x < 0 , | x | = -x
-
According to the definition of the absolute value above,
x > - 2 (given above) is equivalent to x + 2 > 0
if x + 2 > 0 then | x + 2 | = x + 2
-
the above definition gives
| 6 - 9 | = | - 3 | = 3
-
the whole expression given above can now
be written as
2(x + 2) - 3x - (-2 - x) + 3
-
multiply factor
2x + 4 -3x + 2 + x + 3
-
group like terms
9
Matched Exercise 3: If x
> 3, simplify the expression
2| x - 3 | + 6x - (2 - 3x) + | 9 - 20 |
solution
Example 4: Find the slope and the y-intercept of the line given by the equation
2y - 3x = 10
Solution to Example 4
-
We first write the equation in slope
intercept form y = mx +b. Put terms in x and constant terms on the right side
2y = 3x + 10
-
Divide both sides by 2
y = (3/2)x + 5
-
Now that the equation is in slope intercept form, we identify the slope as the coefficient of x and is equal to 3/2 and the y intercept as (0 , 5).
Matched Exercise 4: Find
the slope and the y-intercept of the line given by the equation
-3y - 6x = 7
solution
Example 5: Find the equation of the line
passing through the points (2 , 3) and (4 , 1).
Solution to Example 5
-
We first calculate the slope m
m = (1 - 3) / (4 - 2) = -1
-
We now use the point-slope form of a line to find the equation of the line
y - y1 = m(x - x1) , where m is the slope and (x1,y1) is any of the two points given above.
-
Substitute m by its value -1 and x1 and y1 by 2 and 3 respectively, we obtain the equation of the line.
y - 3 = -1(x - 2)
-
in slope intercept form the equation is written as
y = -x + 5
Matched Exercise 5: Find
the equation of the line passing through the points (0 , 3) and (-1 , 1).
solution
More links and references to pages with algebra problems, tutorials and self tests.