{"id":3731,"date":"2013-06-26T09:29:49","date_gmt":"2013-06-26T16:29:49","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=3731"},"modified":"2020-01-15T10:49:43","modified_gmt":"2020-01-15T18:49:43","slug":"gmat-quant-three-equations-three-unknowns","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/gmat-quant-three-equations-three-unknowns\/","title":{"rendered":"GMAT Quant: Three Equations, Three Unknowns"},"content":{"rendered":"<p>First, here are two challenging <a href=\"https:\/\/magoosh.com\/gmat\/three-algebra-formulas-essential-for-the-gmat\/\" rel=\"noopener noreferrer\" target=\"_blank\">GMAT algebra<\/a> problems, both involving three equations with three algebraic unknowns.<\/p>\n<p>2a + b + 3c = 6<\/p>\n<p>a \u2013 b + 5c = 12<\/p>\n<p>3a + 2b + 2c = 2<\/p>\n<p>1) Given the equations above, what does the product a*b*c equal?<\/p>\n<ul>\n\t(A) 6<br \/>\n\t(B) -6<br \/>\n\t(C) 12<br \/>\n\t(D) -12<br \/>\n\t(E) 24\n<\/ul>\n<p>6x \u2013 5y + 3z = 23<\/p>\n<p>4x + 8y \u2013 11z = 7<\/p>\n<p>5x \u2013 6y + 2z = 12<\/p>\n<p>2) Given the equations above, x + y + z = ?<\/p>\n<ul>\n\t(A) 11<br \/>\n\t(B) 12<br \/>\n\t(C) 13<br \/>\n\t(D) 14<br \/>\n\t(E) 15\n<\/ul>\n<p>Full solutions will follow this article.<\/p>\n<p>&nbsp;<\/p>\n<h2>Simplifying<\/h2>\n<p>In a previous post, I discussed solving <a href=\"https:\/\/magoosh.com\/gmat\/gmat-quant-how-to-solve-two-equations-with-two-variables\/\">two algebra equations with two unknowns<\/a>.\u00a0 That&#8217;s already a challenging task.\u00a0 Three equations with three unknowns is even trickier, and something you are quite unlikely to see unless you are already performing brilliantly on the Quant section.<\/p>\n<p>Well, it turns out, solving these, we use a time-honored problem-solving strategy: <b><i>inside every big problem is a little problem struggling to get out<\/i><\/b>. Yes, that&#8217;s playfully stated, but I have found it is often surprisingly apt in all kinds of personal and interpersonal situations.<\/p>\n<p>What&#8217;s considerably more pertinent here &#8212; this basic idea is the core of much advanced mathematical thinking, at the levels of calculus, analysis, number theory, and other more abstruse topics.\u00a0At all levels, mathematicians strive to reduce problems which they <i>don&#8217;t<\/i> know how to solve to problems which they <i>do<\/i> know how to solve.\u00a0 That can be a hugely valuable perspective on GMAT mathematical strategy.\u00a0 Among other things, that&#8217;s precisely the approach with these problems. I will assume you have read that previous post and are somewhat competent in the two variable\/two equation problems &#8212; for the purposes of this discussion, I will consider those the problems we <i>do<\/i> know how to solve. I will also assume you are familiar with substitution and elimination from that previous post.<\/p>\n<p>&nbsp;<\/p>\n<h2>The approach<\/h2>\n<p>Here&#8217;s the general strategy for solving three equations with three unknowns.<\/p>\n<p><span style=\"text-decoration: underline\">Step #1<\/span>: Pick a pair of equations, two of the three, and using either substitution or elimination, eliminate one of the variables.\u00a0 Most often, elimination is much much easier than substitution!\u00a0 After this step, we will end up with one equation with two unknowns.<\/p>\n<p>For this one, you have to step back and have your right-brain pattern matching hat on.\u00a0 You have to think very strategically about what would be the most efficient.\u00a0 For example, in problem #1 above, if I wanted to pick the first two equations and eliminate c, I <i>could<\/i> do that, but it would involve multiplying the first equation by 5 and the second equation by (-3), which would lead to some big numbers.\u00a0 Hmmm.\u00a0 Not the slickest approach.\u00a0 Instead, when I look at these equations I notice &#8212;- the first equation has a (+b) and the second equation has (-b), so without any fuss, I could add those equations and right away eliminate b.\u00a0 That&#8217;s a considerably more efficient approach.<\/p>\n<p><span style=\"text-decoration: underline\">Step #2<\/span>: Pick a <i>different<\/i> pair of equations, and through elimination, eliminate the <i>same<\/i> variable you eliminated in step #1.\u00a0 As the result of this step, we now will have two equations with the same two unknowns.<\/p>\n<p><span style=\"text-decoration: underline\">Step #3<\/span>: At this point, we have reduced the problem we didn&#8217;t know how to solve to one we do know how to solve: two equations with two unknowns.\u00a0 Use those techniques to solve for those two variables.<\/p>\n<p><span style=\"text-decoration: underline\">Step #4<\/span>: Once you have the numerical values of two of the variable, plug into any of the original equations to solve for the value of the third variable.<\/p>\n<p>I will demonstrate this entire strategy in the solution to #1 below.<\/p>\n<p>&nbsp;<\/p>\n<h2>Do I have to solve?<\/h2>\n<p>For problem #1 above, we have to solve fully, but the coefficients are reasonably small, and as it turns out, the numbers come out nice and neat.\u00a0 By contrast, #2 is a monster.\u00a0 The numbers are larger and uglier, and the answer will come up as ugly fractions.\u00a0\u00a0 But, as it turns out, we can answer the questions being asked with only a minimum of calculations.<\/p>\n<p>This is where you really have to have your <a href=\"https:\/\/magoosh.com\/gmat\/how-to-do-gmat-math-faster\/\">creative, out-of-the-box, right brain<\/a> cap on.\u00a0\u00a0 Question #2 is not asking for the values of individual variables, but for an expression.\u00a0 As it turns out, there&#8217;s an unbelievably simple way to jump directly to the answer with astonishingly little work.\u00a0 Do you see it?\u00a0 I will discuss this in the solution below.<\/p>\n<p>The Moral: Don&#8217;t automatically assume you always have to slog through the hard work of solving for all the individual variables.\u00a0\u00a0 Always keep your antennae up for creative, time-saving shortcuts!<\/p>\n<p>&nbsp;<\/p>\n<h2>Summary<\/h2>\n<p>Once again, these problems are very rare.\u00a0\u00a0 You will not see them at all unless you are performing at 700+ level, getting almost everything else right.\u00a0\u00a0 Here&#8217;s another problem, for further practice:<\/p>\n<p>3) http:\/\/gmat.magoosh.com\/questions\/1009<\/p>\n<p>If you have any questions on what I&#8217;ve said here, let me know in the comments sections below!<\/p>\n<p>&nbsp;<\/p>\n<h2>Solutions to practice problems<\/h2>\n<p>1) Here, I will show the full solution outlined above.\u00a0 First of all, here are the equations, with letter designations.<\/p>\n<p>(<b>P<\/b>) 2a + b + 3c = 6<\/p>\n<p>(<b>Q<\/b>) a \u2013 b + 5c = 12<\/p>\n<p>(<b>R<\/b>) 3a + 2b + 2c = 2<\/p>\n<p>(I started later in the alphabet, so these letter-names of the equations wouldn&#8217;t be confused with answer choice letters!)\u00a0 First of all, I notice that lovely (+b) in (<b>P<\/b>) and (-b) in (<b>Q<\/b>), so I will add those two.<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2013\/06\/tetu_img1.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-3732\" alt=\"tetu_img1\" src=\"https:\/\/magoosh.com\/gmat\/files\/2013\/06\/tetu_img1.png\" width=\"228\" height=\"77\" \/><\/a><\/p>\n<p>That is new equation (<b>S<\/b>), with variable a &amp; c.\u00a0 Now, in step #2, we want to pick a different pair of equations, and eliminate the same variable, b.\u00a0 Again, I like the (-b) in (<b>Q<\/b>) &#8212; that won&#8217;t be hard to use to cancel (+2b) in equation in (<b>R<\/b>).\u00a0 Just multiply (<b>Q<\/b>) by 2, and add it to (<b>R<\/b>).<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2013\/06\/tetu_img2.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-3733\" alt=\"tetu_img2\" src=\"https:\/\/magoosh.com\/gmat\/files\/2013\/06\/tetu_img2.png\" width=\"260\" height=\"77\" \/><\/a><\/p>\n<p>Now, with (<b>S<\/b>) and (<b>T<\/b>), we have two equations with the same two unknowns, a &amp; c.\u00a0 The numbers 8 &amp; 24, the coefficients of c, have a LCM of 24.\u00a0 Multiply (<b>S<\/b>) by 3 and (<b>T<\/b>) by (-2).<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2013\/06\/tetu_img3.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-3734\" alt=\"tetu_img3\" src=\"https:\/\/magoosh.com\/gmat\/files\/2013\/06\/tetu_img3.png\" width=\"238\" height=\"81\" \/><\/a><\/p>\n<p>Thus, a\u00a0 = -2.\u00a0 Plug this value into (<b>S<\/b>).<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2013\/06\/tetu_img4.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-3735\" alt=\"tetu_img4\" src=\"https:\/\/magoosh.com\/gmat\/files\/2013\/06\/tetu_img4.png\" width=\"241\" height=\"50\" \/><\/a><\/p>\n<p>Plug these two values into (<b>P<\/b>).<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2013\/06\/tetu_img5.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-3736\" alt=\"tetu_img5\" src=\"https:\/\/magoosh.com\/gmat\/files\/2013\/06\/tetu_img5.png\" width=\"327\" height=\"50\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2013\/06\/tetu_img5.png 327w, https:\/\/magoosh.com\/gmat\/files\/2013\/06\/tetu_img5-300x45.png 300w\" sizes=\"(max-width: 327px) 100vw, 327px\" \/><\/a><\/p>\n<p>Thus, {a, b, c} = (-2, 1, 3), and their product a*b*c = -6.\u00a0 Answer = (<b>B<\/b>).<\/p>\n<p>2) As with the last problem, I will begin by giving letter names to the equations.<\/p>\n<p>(<b>P<\/b>) 6x \u2013 5y + 3z = 23<\/p>\n<p>(<b>Q<\/b>) 4x + 8y \u2013 11z = 7<\/p>\n<p>(<b>R<\/b>) 5x \u2013 6y + 2z = 12<\/p>\n<p>Here, it would be a colossal waste of time to solve for the individual values of x &amp; y &amp; z separately.\u00a0 We want to find the value of x + y + z.\u00a0 Notice, first of all, that the x-coefficient of (<b>R<\/b>) is one higher than that of (<b>Q<\/b>); unfortunately, their y-coefficients are 14 units apart from each other, not close at all.\u00a0 Now, notice that the x-coefficient of (<b>P<\/b>) is one higher than that of (<b>R<\/b>); also, the y-coefficient of (<b>P<\/b>) is one higher than that of (<b>R<\/b>); also, the z-coefficient of (<b>P<\/b>) is one higher than that of (<b>R<\/b>)!\u00a0 BINGO!\u00a0 The difference, (<b>P<\/b>) \u2013 (<b>R<\/b>), equals the expression we seek!<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2013\/06\/tetu_img6.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-3737\" alt=\"tetu_img6\" src=\"https:\/\/magoosh.com\/gmat\/files\/2013\/06\/tetu_img6.png\" width=\"249\" height=\"76\" \/><\/a><\/p>\n<p>Answer = <b>A<\/b><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>First, here are two challenging GMAT algebra problems, both involving three equations with three algebraic unknowns. 2a + b + 3c = 6 a \u2013 b + 5c = 12 3a + 2b + 2c = 2 1) Given the equations above, what does the product a*b*c equal? (A) 6 (B) -6 (C) 12 (D) [&hellip;]<\/p>\n","protected":false},"author":26,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[138],"tags":[],"ppma_author":[13209],"class_list":["post-3731","post","type-post","status-publish","format-standard","hentry","category-algebra"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v21.7 (Yoast SEO v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>GMAT Quant: Three Equations, Three Unknowns<\/title>\n<meta name=\"description\" content=\"GMAT expert Mike McGarry covers the higher level math concept of three equations with three unknowns\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/magoosh.com\/gmat\/gmat-quant-three-equations-three-unknowns\/\" \/>\n<meta property=\"og:locale\" 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M\u1d9cGarry","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/magoosh.com\/gmat\/#\/schema\/person\/image\/15a1e36ef1c2c3940179212433de141a","url":"https:\/\/secure.gravatar.com\/avatar\/6b06de81592cd77bb46aa560cc59aee179cba4d042835c3529221ea1b344cce0?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/6b06de81592cd77bb46aa560cc59aee179cba4d042835c3529221ea1b344cce0?s=96&d=mm&r=g","caption":"Mike M\u1d9cGarry"},"description":"Mike holds an A.B. in Physics (graduating magna cum laude) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's Youtube video explanations.","sameAs":["https:\/\/www.youtube.com\/c\/MagooshGMATChannel\/featured"],"award":["Magna cum laude from Harvard"],"knowsAbout":["GMAT"],"knowsLanguage":["English"],"jobTitle":"Content Creator","worksFor":"Magoosh","url":"https:\/\/magoosh.com\/gmat\/author\/mikemcgarry\/"}]}},"authors":[{"term_id":13209,"user_id":26,"is_guest":0,"slug":"mikemcgarry","display_name":"Mike M\u1d9cGarry","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/6b06de81592cd77bb46aa560cc59aee179cba4d042835c3529221ea1b344cce0?s=96&d=mm&r=g","user_url":"","last_name":"M\u1d9cGarry","first_name":"Mike","description":"Mike served as a GMAT Expert at Magoosh, helping create hundreds of lesson videos and practice questions to help guide GMAT students to success. He was also featured as \"member of the month\" for over two years at <a href=\"https:\/\/gmatclub.com\/blog\/2012\/09\/mike-mcgarrys-gmat-experience\/\" rel=\"noopener noreferrer\">GMAT Club<\/a>. Mike holds an A.B. in Physics (graduating <em>magna cum laude<\/em>) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's <a href=\"https:\/\/www.youtube.com\/c\/MagooshGMATChannel\/featured\" rel=\"noopener noreferrer\">Youtube <\/a>video explanations and resources like <a href=\"https:\/\/magoosh.com\/gmat\/whats-a-good-gmat-score\/\" rel=\"noopener noreferrer\">What is a Good GMAT Score?<\/a> and the <a href=\"https:\/\/magoosh.com\/gmat\/gmat-diagnostic-test\/\" rel=\"noopener noreferrer\">GMAT Diagnostic Test<\/a>."}],"_links":{"self":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/3731","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/users\/26"}],"replies":[{"embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/comments?post=3731"}],"version-history":[{"count":0,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/3731\/revisions"}],"wp:attachment":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/media?parent=3731"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/categories?post=3731"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/tags?post=3731"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/ppma_author?post=3731"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}