# 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