{"id":1261,"date":"2023-04-04T19:08:38","date_gmt":"2023-04-04T19:08:38","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=1261"},"modified":"2024-10-18T20:51:09","modified_gmt":"2024-10-18T20:51:09","slug":"algebraic-equations-learn-it-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/algebraic-equations-learn-it-2\/","title":{"raw":"Algebraic Equations: Learn It 2","rendered":"Algebraic Equations: Learn It 2"},"content":{"raw":"

Formulas<\/h2>\r\n

An equation<\/strong> is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation [latex]2x+1=7[\/latex] has the unique solution [latex]x=3[\/latex] because when we substitute [latex]3[\/latex] for [latex]x[\/latex] in the equation, we obtain the true statement [latex]2\\left(3\\right)+1=7[\/latex].<\/p>\r\n

A formula<\/strong> is an equation expressing a relationship between constant and variable quantities. Very often the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area [latex]A[\/latex] of a circle in terms of the radius [latex]r[\/latex] of the circle: [latex]A=\\pi {r}^{2}[\/latex]. For any value of [latex]r[\/latex], the area [latex]A[\/latex] can be found by evaluating the expression [latex]\\pi {r}^{2}[\/latex].<\/p>\r\n

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Equations and formulas<\/h3>\r\n
    \r\n\t
  • An equation<\/strong> is a mathematical statement that shows the equality of two expressions, typically separated by an equal sign. It states that the two expressions have the same value, and the values of variables that make the equation true are called solutions.<\/li>\r\n\t
  • A formula<\/strong> is a mathematical expression that represents a relationship or a rule between variables or quantities. It usually contains variables, constants, and arithmetic operations, and is used to calculate or derive a particular result or value.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>\r\n
    A right circular cylinder with radius [latex]r[\/latex] and height [latex]h[\/latex] has the surface area [latex]S[\/latex] (in square units) given by the formula [latex]S=2\\pi r\\left(r+h\\right)[\/latex]. Find the surface area of a cylinder with radius [latex]6[\/latex] in. and height [latex]9[\/latex] in. Leave the answer in terms of [latex]\\pi[\/latex].
    \"A<\/center>
    Right circular cylinder<\/span><\/strong><\/center>\r\n

     <\/p>\r\n

    [reveal-answer q=\"257174\"]Show Solution[\/reveal-answer]
    \r\n[hidden-answer a=\"257174\"]<\/p>\r\n

    Evaluate the expression [latex]2\\pi r\\left(r+h\\right)[\/latex] for [latex]r=6[\/latex] and [latex]h=9[\/latex].<\/p>\r\n

    [latex]\\begin{align}S&=2\\pi r\\left(r+h\\right) \\\\ & =2\\pi\\left(6\\right)[\\left(6\\right)+\\left(9\\right)] \\\\ & =2\\pi\\left(6\\right)\\left(15\\right) \\\\ & =180\\pi\\end{align}[\/latex]<\/p>\r\n

    The surface area is [latex]180\\pi [\/latex] square inches.<\/p>\r\n

    [\/hidden-answer]<\/p>\r\n<\/section>\r\n

    \"An<\/center>\r\n

     <\/p>\r\n

    A photograph with length [latex]L[\/latex] and width [latex]W[\/latex] is placed in a mat of width [latex]8[\/latex] centimeters (cm). The area of the mat (in square centimeters, or cm[latex]2[\/latex]<\/sup>) is found to be [latex]A=\\left(L+16\\right)\\left(W+16\\right)-L\\cdot W[\/latex]. Find the area of a mat for a photograph with length [latex]32[\/latex] cm and width [latex]24[\/latex] cm.[reveal-answer q=\"846181\"]Show Solution[\/reveal-answer]
    \r\n[hidden-answer a=\"846181\"][latex]1,152 cm^{2}[\/latex][\/hidden-answer]<\/p>\r\n<\/section>\r\n

    Simplify Algebraic Expressions<\/h2>\r\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>\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.
    \r\n
    \r\nTo multiply fractions<\/strong>, multiply the numerators and place them over the product of the denominators.\r\n\r\n\r\n

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

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

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

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

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

    Simplify the following algebraic expressions:\r\n\r\n\r\n
      \r\n\t
    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\"]<\/p>\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]<\/center><\/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]<\/center><\/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]<\/center><\/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]<\/center><\/li>\r\n<\/ol>\r\n

        [\/hidden-answer]<\/p>\r\n<\/section>\r\n

        [ohm2_question hide_question_numbers=1]6759[\/ohm2_question]<\/section>","rendered":"

        Formulas<\/h2>\n

        An equation<\/strong> is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation [latex]2x+1=7[\/latex] has the unique solution [latex]x=3[\/latex] because when we substitute [latex]3[\/latex] for [latex]x[\/latex] in the equation, we obtain the true statement [latex]2\\left(3\\right)+1=7[\/latex].<\/p>\n

        A formula<\/strong> is an equation expressing a relationship between constant and variable quantities. Very often the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area [latex]A[\/latex] of a circle in terms of the radius [latex]r[\/latex] of the circle: [latex]A=\\pi {r}^{2}[\/latex]. For any value of [latex]r[\/latex], the area [latex]A[\/latex] can be found by evaluating the expression [latex]\\pi {r}^{2}[\/latex].<\/p>\n

        \n
        \n

        Equations and formulas<\/h3>\n
          \n
        • An equation<\/strong> is a mathematical statement that shows the equality of two expressions, typically separated by an equal sign. It states that the two expressions have the same value, and the values of variables that make the equation true are called solutions.<\/li>\n
        • A formula<\/strong> is a mathematical expression that represents a relationship or a rule between variables or quantities. It usually contains variables, constants, and arithmetic operations, and is used to calculate or derive a particular result or value.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n
          A right circular cylinder with radius [latex]r[\/latex] and height [latex]h[\/latex] has the surface area [latex]S[\/latex] (in square units) given by the formula [latex]S=2\\pi r\\left(r+h\\right)[\/latex]. Find the surface area of a cylinder with radius [latex]6[\/latex] in. and height [latex]9[\/latex] in. Leave the answer in terms of [latex]\\pi[\/latex].<\/p>\n
          \"A<\/div>\n
          Right circular cylinder<\/span><\/strong><\/div>\n

           <\/p>\n