
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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