Solutions to Algebra Problems

Detailed solutions to algebra problems are presented.
Solution to Problem 1: Given the equation
5(-3x - 2) - (x - 3) = -4(4x + 5) + 13
Multiply factors.
-15x - 10 - x + 3 = -16x - 20 +13
Group like terms.
-16x - 7 = -16x - 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) + 4b - 2(a -b -3) + 5
Multiply factors.
= 2a - 6 + 4b -2a + 2b + 6 + 5
Group like terms.
= 6b + 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
2x - 4y = 9
To find the x intercept we set y = 0 and solve for x.
2x - 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) = 6x + 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 = mx + 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.
|-2x + 2| -3 = -3
Add 3 to both sides of the equation and simplify.
|-2x + 2| = 0
|-2x + 2| is equal to 0 if -2x + 2 = 0. Solve for x to obtain
x = 1

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