{"id":522,"date":"2024-04-22T20:25:31","date_gmt":"2024-04-22T20:25:31","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=522"},"modified":"2024-11-20T00:52:12","modified_gmt":"2024-11-20T00:52:12","slug":"introduction-to-real-numbers-learn-it-6","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/introduction-to-real-numbers-learn-it-6\/","title":{"raw":"Introduction to Real Numbers: Learn It 6","rendered":"Introduction to Real Numbers: Learn It 6"},"content":{"raw":"

Simplifying Algebraic Expressions<\/h2>\r\n
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\r\n\r\nSometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.\r\n\r\n
When simplifying algebraic expressions, we may sometimes need to add, subtract, simplify, multiply, or divide fractions. It is important to be able to do these operations on the fractions without converting them to decimals.To multiply fractions<\/strong>, multiply the numerators and place them over the product of the denominators.\r\n

[latex]\\dfrac{a}{b}\\cdot\\dfrac{c}{d} = \\dfrac {ac}{bd}[\/latex]<\/p>\r\nTo divide fractions<\/strong>, multiply the first by the reciprocal of the second.\r\n

\u00a0[latex]\\dfrac{a}{b}\\div\\dfrac{c}{d}=\\dfrac{a}{b}\\cdot\\dfrac{d}{c}=\\dfrac{ad}{bc}[\/latex]<\/p>\r\nTo simplify fractions<\/strong>, find common factors in the numerator and denominator that cancel.\r\n

\u00a0[latex]\\dfrac{24}{32}=\\dfrac{2\\cdot2\\cdot2\\cdot3}{2\\cdot2\\cdot2\\cdot2\\cdot2}=\\dfrac{3}{2\\cdot2}=\\dfrac{3}{4}[\/latex]<\/p>\r\nTo add or subtract fractions<\/strong>, first rewrite each fraction as an equivalent fraction such that each has a common denominator, then add or subtract the numerators and place the result over the common denominator.\r\n

\u00a0[latex]\\dfrac{a}{b}\\pm\\dfrac{c}{d} = \\dfrac{ad \\pm bc}{bd}[\/latex]<\/p>\r\n\r\n<\/section>

Simplify each algebraic expression.\r\n
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  1. [latex]3x - 2y+x - 3y - 7[\/latex]<\/li>\r\n \t
  2. [latex]2r - 5\\left(3-r\\right)+4[\/latex]<\/li>\r\n \t
  3. [latex]\\left(4t-\\dfrac{5}{4}s\\right)-\\left(\\dfrac{2}{3}t+2s\\right)[\/latex]<\/li>\r\n \t
  4. [latex]2mn - 5m+3mn+n[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"286046\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"286046\"]\r\n
      \r\n \t
    1. [latex]\\begin{align}3x-2y+x-3y-7 & =3x+x-2y-3y-7 && \\text{Commutative property of addition} \\\\ & =4x-5y-7 && \\text{Simplify} \\\\ \\text{ }\\end{align}[\/latex]<\/li>\r\n \t
    2. [latex]\\begin{align}2r-5\\left(3-r\\right)+4 & =2r-15+5r+4 && \\text{Distributive property}\\\\&=2r+5r-15+4 && \\text{Commutative property of addition} \\\\ & =7r-11 && \\text{Simplify} \\\\ \\text{ }\\end{align}[\/latex]<\/li>\r\n \t
    3. [latex]\\begin{align} 4t-\\frac{5}{4}s -\\left(\\frac{2}{3}t+2s\\right) &=4t-\\frac{5}{4}s-\\frac{2}{3}t-2s &&\\text{Distributive property}\\\\&=4t-\\frac{2}{3}t-\\frac{5}{4}s-2s && \\text{Commutative property of addition}\\\\&=\\frac{12}{3}t-\\frac{2}{3}t-\\frac{5}{4}s-\\frac{8}{4}s && \\text{Common Denominators}\\\\ & =\\frac{10}{3}t-\\frac{13}{4}s && \\text{Simplify} \\\\ \\text{ }\\end{align}[\/latex]<\/li>\r\n \t
    4. [latex]\\begin{align}mn-5m+3mn+n & =2mn+3mn-5m+n && \\text{Commutative property of addition} \\\\ & =5mn-5m+n && \\text{Simplify}\\end{align}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
      [ohm2_question hide_question_numbers=1]18718[\/ohm2_question]<\/section>
      [ohm2_question hide_question_numbers=1]18719[\/ohm2_question]<\/section>
      [ohm2_question hide_question_numbers=1]18720[\/ohm2_question]<\/section>Now that we've gained a good handle on simplifying algebraic equations by combining like terms and using the order of operations effectively, let's broaden our skills to include simplifying formulas. Just like with equations, simplifying formulas helps us to see the underlying structure more clearly and makes them easier to use.\r\n\r\n
      A rectangle with length [latex]L[\/latex] and width [latex]W[\/latex] has a perimeter [latex]P[\/latex] given by [latex]P=L+W+L+W[\/latex]. Simplify this expression.[reveal-answer q=\"921194\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"921194\"]\r\n
      [latex]\\begin{align}&P=L+W+L+W \\\\ &P=L+L+W+W && \\text{Commutative property of addition} \\\\ &P=2L+2W && \\text{Simplify} \\\\ &P=2\\left(L+W\\right) && \\text{Distributive property}\\end{align}[\/latex]<\/div>\r\n
      [\/hidden-answer]<\/div>\r\n<\/section>Whether it's a formula for calculating the area of a shape, the interest on an investment, or the speed of an object, making these formulas simpler can make our calculations quicker and our results easier to understand and apply in practical situations.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

      Simplifying Algebraic Expressions<\/h2>\n
      \n
      \n
      \n

      Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.<\/p>\n

      When simplifying algebraic expressions, we may sometimes need to add, subtract, simplify, multiply, or divide fractions. It is important to be able to do these operations on the fractions without converting them to decimals.To multiply fractions<\/strong>, multiply the numerators and place them over the product of the denominators.<\/p>\n

      [latex]\\dfrac{a}{b}\\cdot\\dfrac{c}{d} = \\dfrac {ac}{bd}[\/latex]<\/p>\n

      To divide fractions<\/strong>, multiply the first by the reciprocal of the second.<\/p>\n

      \u00a0[latex]\\dfrac{a}{b}\\div\\dfrac{c}{d}=\\dfrac{a}{b}\\cdot\\dfrac{d}{c}=\\dfrac{ad}{bc}[\/latex]<\/p>\n

      To simplify fractions<\/strong>, find common factors in the numerator and denominator that cancel.<\/p>\n

      \u00a0[latex]\\dfrac{24}{32}=\\dfrac{2\\cdot2\\cdot2\\cdot3}{2\\cdot2\\cdot2\\cdot2\\cdot2}=\\dfrac{3}{2\\cdot2}=\\dfrac{3}{4}[\/latex]<\/p>\n

      To add or subtract fractions<\/strong>, first rewrite each fraction as an equivalent fraction such that each has a common denominator, then add or subtract the numerators and place the result over the common denominator.<\/p>\n

      \u00a0[latex]\\dfrac{a}{b}\\pm\\dfrac{c}{d} = \\dfrac{ad \\pm bc}{bd}[\/latex]<\/p>\n<\/section>\n

      Simplify each algebraic expression.<\/p>\n
        \n
      1. [latex]3x - 2y+x - 3y - 7[\/latex]<\/li>\n
      2. [latex]2r - 5\\left(3-r\\right)+4[\/latex]<\/li>\n
      3. [latex]\\left(4t-\\dfrac{5}{4}s\\right)-\\left(\\dfrac{2}{3}t+2s\\right)[\/latex]<\/li>\n
      4. [latex]2mn - 5m+3mn+n[\/latex]<\/li>\n<\/ol>\n