Solutions to Algebra Problems

Detailed solutions to algebra problems are presented.
Solution to Problem 1:
Given the equation
5 (- 3 x - 2) - (x - 3) = - 4 (4 x + 5) + 13
Multiply factors.
-15 x - 10 - x + 3 = - 16 x - 20 + 13
Group like terms.
- 16 x - 7 = - 16 x - 7
Add 16x + 7 to both sides and write the equation as follows
0 = 0
The above statement is true for all values of x and therefore all real numbers are solutions to the given equation.

Solution to Problem 2:Given the algebraic expression
2 (a -3) + 4 b - 2 (a - b - 3) + 5
Multiply factors.
= 2 a - 6 + 4 b - 2 a + 2 b + 6 + 5
Group like terms.
= 6 b + 5

Solution to Problem 3:Given the expression
| x - 2 | - 4 | -6 |
If x < 2 then x - 2 < 0 and if x - 2 < 0 then |x - 2| = - (x - 2).
Substitute |x - 2| by - (x - 2) and | - 6 | by 6
|x - 2| - 4| -6 | = - (x - 2) - 4(6) = - x - 22

Solution to Problem 4:The distance d between points (-4 , -5) and (-1 , -1) is given by
d = √[ (-1 - (-4))
2 + (-1 - (-5)) 2 ]
Simplify.
d = √(9 + 16) = 5


Solution to Problem 5:Given the equation
2 x - 4 y = 9
To find the x intercept we set y = 0 and solve for x.
2 x - 0 = 9
Solve for x.
x = 9 / 2
The x intercept is at the point (9/2 , 0).

Solution to Problem 6:Given the function
f(x) = 6 x + 1
f(2) - f(1) is given by.
f(2) - f(1) = (6 2 + 1) - (6 1 + 1) = 6

Solution to Problem 7:Given the points (-1, -1) and (2 , 2), the slope m is given by
m = (y2 - y1) / (x2 - x1) = (2 - (-1)) / (2 - (-1)) = 1

Solution to Problem 8:Given the line
5x - 5y = 7
Rewrite the equation in slope intercept form y = m x + b and identify the value of m the slope.
- 5y = - 5x + 7
y = x - 7/5
The slope is given by the coefficient of x which is 1.

Solution to Problem 9:To find the equation of the line through the points (-1 , -1) and (-1 , 2), we first use the slope m.
m = (y2 - y1) / (x2 - x1) = (2 - (-1)) / (-1 - (-1)) = 3 / 0
The slope is undefined which means the line is perpendicular to the x axis and its equation has the form x = constant. Since both points have equal x coordinates -1, the equation is given by:
x = -1

Solution to Problem 10:The equation to solve is given by.
|-2 x + 2| -3 = -3
Add 3 to both sides of the equation and simplify.
|-2 x + 2| = 0
|-2 x + 2| is equal to 0 if -2 x + 2 = 0. Solve for x to obtain
x = 1

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