Lesson HOW TO algebreze and to solve word problems on numbers?
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<H2>How to algebreze and to solve word problems on numbers?</H2> This lesson teaches you, by examples, how to write the expressions and the equations for word problems on numbers. <H3>Problem 1</H3>Laura is thinking of a number such that the sum of the number and five times two more than the number is 26 more than four times the number. Determine the number Laura is thinking of. <B>Solution</B> Let <B>x</B> be the unknown number. Then "the sum of the number and five times two more than the number" is x + 5*(x+2). "26 more than four times the number" is 4x + 26. Thus the equation is x + 5*(x+2) = 4x + 26. Solve it: x + 5x + 10 = 4x + 26. 6x - 4x = 26 - 10, 2x = 16, x = {{{16/2}}} = 8. <B>Answer</B>. Laura is thinking of a number 8. <H3>Problem 2</H3>Twice the differences between a number, x, and fifteen is three times the number. Find the number. <B>Solution</B> "Twice the differences between a number, x, and fifteen" is 2*(x-15). "Three times the number" is 3x. Therefore, you have an equation 2*(x-15) = 3x. 2x - 30 = 3x, 3x - 2x = -30, x = -30. <B>Answer</B>. The number is -30. <H3>Problem 3</H3>Write an algebraic expression for this sentence: ten more than twice the absolute value of a number. <B>Solution</B> 2*|x| + 10. <H3>Problem 4</H3>Write an algebraic equation for this sentence: 14 less than 3 times a number is 5 more than half of the number. <B>Solution</B> "14 less than 3 times a number" is 3x - 14. "5 more than half of the number" is {{{x/2}}} + 5. Therefore, an equation is 3x - 14 = {{{x/2}}} + 5. <H3>Problem 5</H3>Write an algebraic expression for this sentence: the sum of seven more than twice a number and the number. <B>Solution</B> (2x + 7) + x <H3>Problem 6</H3>Write an algebraic equation for this sentence: three times the sum of a number and 3 is the same as five times the number divided by 7. <B>Solution</B> 3*(x+3) = {{{5x/7}}}. <H3>Problem 7</H3>Write an algebraic equation for this sentence: the sum of a number and 3, multiplied by the sum of the number and 8 is 19. <B>Solution</B> (x + 3)*(x + 8) = 19. <H3>Problem 8</H3>Write an algebraic equation for this sentence: three times a number added to 4 times a number is the same as six times the number minus 6 <B>Solution</B> 3x + 4x = 6x - 6. <H3>Problem 9</H3>Write an algebraic equation for this sentence and solve it: the difference of five times a number and 5 results in twice the difference of three times the number and 3. <B>Solution</B> According to the condition, an equation for the unknown number <B>x</B> is 5x - 5 = 2*(3x - 3). Solve it: 5x - 5 = 6x - 6 5x - 6x = -6 + 5, -x = -1, x = 1. <B>Answer</B>. x = 1. <H3>Problem 10</H3>Five times the sum of twice a number and 5 is the same as the sum of 9 times the number and 5. Find the number. <B>Solution</B> The equation is 5*(2x + 5) = 9x + 5. Solve it: 10x + 25 = 9x + 5, 10x - 9x = 5-25 = -20. x = -20. <U>Check</U>: The left side is 5*(2x + 5) = 5*(2*(-20) + 5) = 5*(-40 + 5) = 5*(-35) = -175. The right side is 9x + 5 = 9*(-20) + 5 = -180 + 5 = -175. <B>Answer</B>. x = -20. <H3>Problem 11</H3>The sum of two numbers is 90. The larger number is 14 more than 3 times the smaller number. Find the numbers. <B>Solution</B> Let x be the smaller number. The the other number is 3x + 14. We know that their sum is 90. It gives an equation x + (3x + 14) = 90. Solve it step by step: x + 3x + 14 = 90, 4x = 90 - 14, 4x = 76, x = {{{76/4}}} = 19. It is the smaller number. The other number is 90 - 19 = 71. (it is really in 14 more than 3x: 3*19 + 14 = 57 +14 = 71). 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