{"id":12448,"date":"2020-07-18T03:07:11","date_gmt":"2020-07-17T19:07:11","guid":{"rendered":"https:\/\/www.intmath.com\/blog\/?p=12448"},"modified":"2020-07-18T03:07:11","modified_gmt":"2020-07-17T19:07:11","slug":"finding-integers-in-algebraic-equations","status":"publish","type":"post","link":"https:\/\/www.intmath.com\/blog\/mathematics\/finding-integers-in-algebraic-equations-12448","title":{"rendered":"Finding Integers in Algebraic Equations"},"content":{"rendered":"<p><strong>Integers<\/strong>\u00a0include any\u00a0<strong>whole number<\/strong>\u00a0found on a number line. They also include any negative number that falls on a number line. Whole numbers are strictly positive. Integers do not include anything that is a fraction or decimal.<\/p>\n<p>For example:<\/p>\n<ul>\n<li>-4 is an integer.<\/li>\n<li>0 is an integer.<\/li>\n<li>5.3 is not an integer.<\/li>\n<\/ul>\n<p>It is worth noting that some people learn differently\u00a0when it comes to whole numbers.\u00a0<a href=\"https:\/\/www.mathsisfun.com\/whole-numbers.html\">Some<\/a>\u00a0do not believe that 0 is a whole number and that whole numbers automatically include negatives. If this criterion applies to you, simply adjust the definition of an integer provided here.<\/p>\n<p>It\u2019s that easy! It stays that easy when finding them in algebraic equations, too. This is because integers are simply elements added to mathematical equations. Because they are so identifiable, there\u2019s no question when you come across one -- except maybe when it\u2019s hidden in a variable.<\/p>\n<h2>Parts of an Algebraic Equation<\/h2>\n<p>An algebraic equation incorporates more advanced elements than your basic arithmetic equation. Also referred to as algebraic expressions, these make use of variables, integers and other numbers, and at least one algebraic operation.<\/p>\n<p><strong>Example:<\/strong><\/p>\n<p><em>2x + 3y(15-3) = 30<\/em><\/p>\n<p>The variables in the equation are\u00a0<em>x<\/em>\u00a0and\u00a0<em>y<\/em>.\u00a0<strong>Variables\u00a0<\/strong>stand in place of\u00a0numbers for which we must solve. We do not yet know if the variable is an integer or not. Remember, integers do not include fractions\u00a0and decimals.<\/p>\n<p>The\u00a0<strong>coefficient<\/strong>\u00a0-- the number by which we are multiplying our variable -- for term\u00a0<em>2x<\/em>\u00a0is 2. For\u00a0<em>3y<\/em>\u00a0it is 3.<\/p>\n<p>Our\u00a0<strong>algebraic operations<\/strong>\u00a0include addition (+), subtraction (-), and multiplication (<em>3y(15-3)...<\/em>).<\/p>\n<p>However, we can identify integers in our equation right away. These are 2, 3, 15, and 30.<\/p>\n<h2>Revealing the Variable<\/h2>\n<p>There\u2019s only one way to determine if a variable is an integer or not, and that\u2019s to solve the equation. Not every problem you\u2019ll solve will provide you with variable values. In fact, you\u2019ll most likely be creating your own algebraic equations where you need to solve for a variable in real life. This is especially true if calculating profit. You can use any variable to represent your value.<\/p>\n<p>In this case,\u00a0<em>a=$12,000 (total revenue)<\/em>,<em>\u00a0b= $10,000.45 (total expense)<\/em>, and\u00a0<em>c=profit<\/em>.<\/p>\n<p><em>a - b = c<\/em><\/p>\n<p>Insert your values into the profit equation above.<\/p>\n<p><em>$12,000 - $10,000.45 = c<\/em><\/p>\n<p>Solve for\u00a0<em>c<\/em>.<\/p>\n<p><em>$1,999.55 = c<\/em><\/p>\n<p>In this case,\u00a0<em>c<\/em>\u00a0is not representing an integer. If your values are not whole numbers and include parts -- if there are cents with your dollars -- unless the\u00a0parts negate each other, your profit will not be an integer.<\/p>\n<h3>Solving for a Variable\u2019s Value<\/h3>\n<p>Because we have to solve to find a variable that could be hiding an integer, let\u2019s learn a simple way to solve for the variable. This equation will use like terms with one variable.<\/p>\n<p><em>14x + 13(7*6) - 6x = 610<\/em><\/p>\n<h4>1. Combine Like Terms<\/h4>\n<p>First, let\u2019s combine the like terms in this equation:\u00a0<em>14x\u00a0<\/em>and\u00a0<em>-6x<\/em>. Only when a variable is the same can you combine them.<\/p>\n<p><em>8x + 13(7*6) = 610<\/em><\/p>\n<h4>2. Follow Order of Operations<\/h4>\n<p>Following Order of Operations, we solve what is in parentheses first, followed by multiplication.<\/p>\n<p><em>8x + 13(42) = 610<\/em><\/p>\n<p><em>8x + 546 = 610<\/em><\/p>\n<h4>3. Isolate the Variable<\/h4>\n<p>Next, we need to start isolating the variable or removing the remaining numbers and the coefficient from the left side of the equation. The rule goes that anything done to one side of the equation must also be done to the other. In this case, we will subtract 546 from both sides.<\/p>\n<p><em>8x = 64<\/em><\/p>\n<p>Finally, to fully isolate the variable, divide\u00a0<em>8x<\/em>\u00a0by\u00a0<em>8<\/em>\u00a0and\u00a0<em>64<\/em>\u00a0by\u00a0<em>8<\/em>.<\/p>\n<p><em>x=8<\/em><\/p>\n<h4>4. Prove Your Answer<\/h4>\n<p>According to this answer, our variable is an integer. Let\u2019s prove our equation by placing the variable in the equation and solving for the result, 610.<\/p>\n<p><em>14(8) + 13(7*6) - 6(8) = 610<\/em><\/p>\n<p><em>112 + 13(42) - 48<\/em><\/p>\n<p><em>112 + 546 - 48<\/em><\/p>\n<p><em>658 - 48 = 610<\/em><\/p>\n<p>Our integer works!<\/p>\n<h2>Conclusion<\/h2>\n<p>Finding integers in algebraic equations is all about knowing what is considered an integer or not. It also relies on your ability to solve for variables. Again, never assume variables are integers. There are plenty of cases when they aren\u2019t.<\/p>\n<p class=\"alt\">See the <a href=\"https:\/\/www.intmath.com\/blog\/mathematics\/finding-integers-in-algebraic-equations-12448#comments\" id=\"comms\">1 Comment<\/a> below.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Integers\u00a0include any\u00a0whole number\u00a0found on a number line. They also include any negative number that falls on a number line. Whole numbers are strictly positive. Integers do not include anything that is a fraction or decimal. For example: -4 is an integer. 0 is an integer. 5.3 is not an integer. It is worth noting that [&hellip;]<\/p>\n","protected":false},"author":15,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_mo_disable_npp":""},"categories":[4],"tags":[],"_links":{"self":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts\/12448"}],"collection":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/users\/15"}],"replies":[{"embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/comments?post=12448"}],"version-history":[{"count":1,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts\/12448\/revisions"}],"predecessor-version":[{"id":12449,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts\/12448\/revisions\/12449"}],"wp:attachment":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/media?parent=12448"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/categories?post=12448"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/tags?post=12448"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}