{"id":4353,"date":"2022-11-23T10:00:01","date_gmt":"2022-11-23T04:30:01","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=4353"},"modified":"2022-11-23T10:29:18","modified_gmt":"2022-11-23T04:59:18","slug":"what-is-a-linear-equation","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/what-is-a-linear-equation\/","title":{"rendered":"What is a Linear Equation"},"content":{"rendered":"
Let us consider a small puzzle: Think of a number and add 7 to get 12. What is the number?
\nWe can easily say that the number must be 5.
\nIf we use a letter, i.e., (variable) to stand for the unknown number, we can write this puzzle as follows:
\nx + 7 = 12 \u00a0 \u00a0 \u00a0 \u00a0 …(1)
\nHere, if x = 5, 5 + 7 = 12. So, x = 5 is the unknown number which satisfies the statement (1). This is nothing but an equation. An equation is a mathematical statement equating two expressions. The expression on the left side of the equal sign is called LHS (Left Hand Side) and the expression on the right side of the equal sign is called RHS (Right Hand Side). The expressions on either side of equal sign are called members of the equation.<\/p>\n
An equation which involves a variable with the highest power 1 is called a linear equation<\/strong>. Read More:<\/strong><\/p>\n Example 1:<\/strong> Write the following mathematical statements as algebraic equations: Example 2:<\/strong> Convert the following equations into statements: An equation behaves like a pair of balanced scales. Both sides of the equation are balanced in the same\u00a0manner as the scales of a balance. In order to main-tain balance, we need to remember some rules. To solve an equation means to find a number which when substituted for the variable in the equation, makes its LHS equal to the RHS. This number which satisfies the equation is called the solution<\/strong> or root<\/strong> of the equation. Trial and Error Method<\/strong> Systematic Method<\/strong> Example 1:<\/strong> Solve: 2m – 12 = 18. Example 3:<\/strong> Three times a number decreased by 6 gives 12. Find the number. What is a Linear Equation Let us consider a small puzzle: Think of a number and add 7 to get 12. What is the number? We can easily say that the number must be 5. If we use a letter, i.e., (variable) to stand for the unknown number, we can write this puzzle as follows: […]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[5],"tags":[2409,2408,2412,2411],"yoast_head":"\n
\nExamples:<\/strong> 2x + 3 = 7, x + y = 9, a + b = 2.5 are examples of a linear equation.
\nBut, what about x2<\/sup> + 4 = 13? Is it a linear equation? No because variable x has its power 2.<\/p>\n\n
\n(a) 3 more than 5 times of x gives 18.
\n(b) The sum of p and 7 gives 12.
\n(c) One third of a number increased by 5 is 8.
\nSolution:<\/strong>
\n(a)<\/strong> Five times x = 5x
\n3 more than five times x = 5x + 3.
\nIt is 18.
\n\u2234 The equation is 5x + 3 = 18.
\n(b)<\/strong> Sum ofp and 7 = p + 7.
\nIt is 12.
\n\u2234 The equation is p + 7 = 12
\n(c)<\/strong> Let the number be x.
\nOne-third of x = 1\/3 of x = x\/3
\nOne-third of x increased by 5 = x\/3 + 5.
\nIt is 8.
\n\u2234 The equation is x\/3 + 5 = 8<\/p>\n
\n(a) x + 2 = 5 \u00a0 \u00a0 \u00a0 (b) 5p + 3 = 28 \u00a0 \u00a0 \u00a0 \u00a0 (c) 2\/3 m – 1 = 5
\nSolution:<\/strong>
\n(a) The sum of x and 2 gives 5.
\n(b) 3 more than five times of p gives 28.
\n(c) 1 less than two-third of m is 5.<\/p>\nEQUATIONS WITH A PAIR OF SCALES<\/strong><\/h3>\n
\n Rules for balancing linear equations<\/strong>
\nLet us consider 6 + 2 = 8. It is an equality, because both of its sides (LHS and RHS) equal to 8.
\nRule 1:<\/strong> If we add the same quantity on both sides of an equality, the equality holds true.
\n6 + 2 = 8
\n6 + 2 + 3 = 8 + 3 \u00a0\u00a0(adding 3 on both sides)
\n11 = 11 (LHS = RHS)
\nRule 2:<\/strong> If we subtract the same quantity from both sides of an equality, the equality holds true.
\n6 + 2 = 8
\n6 + 2 – 3 = 8 – 3 \u00a0\u00a0(subtracting 3 from both sides)
\n5 = 5 (LHS = RHS)
\nRule 3:<\/strong> If we multiply both sides of an equality by the same quantity, the equality holds true.
\n6 + 2 = 8
\n(6 + 2) \u00d7\u00a03 = 8 \u00d7\u00a03 \u00a0\u00a0(multiplying by 3 on both sides)
\n6 \u00d7\u00a03\u00a0+ 2 \u00d7\u00a03 = 8 \u00d7\u00a03
\n18 + 6 = 24
\n24 = 24 (LHS = RHS)
\nRule 4:<\/strong> If we divide both sides of an equality by the same quantity, the equality holds true.
\n6 + 2 = 8
\n\\(\\frac{6+2}{3}=\\frac{8}{3}\\) \u00a0 \u00a0(dividing by 3 on both sides)
\n\\(\\frac{8}{3}=\\frac{8}{3}\\) \u00a0 (LHS = RHS)<\/p>\nSOLVING EQUATIONS<\/strong><\/h3>\n
\nExample:<\/strong> 2x + 3 = 11
\nHere, if we consider the value of x = 4,
\nthen 2x + 3 = 11
\ni. e., LHS = RHS
\nso, 4 is the root of 2x + 3 = 11.
\nTo find the root (solution) of an equation, i.e., to solve an equation we can follow these methods:
\n1. Trial and error method
\n2. Systematic method<\/p>\n
\nIn this method, we try different values for the variable to make both sides of an equation equal. We stop this trial as soon as we get a particular value of the variable which makes the LHS equal to the RHS. This particular value is said to be the root of that equation.
\nExample:<\/strong> Solve 2x – 3 = 5, using trial and error method.
\nSolution:<\/strong> We try different values of x to find LHS = RHS.
\n
\nFrom the above table, we find that LHS = RHS, when x = 4.
\n\u2234 Solution is x = 4.<\/p>\n
\nTo solve a linear equation using this method, we add, subtract, multiply, or divide both the sides of the equation by the same number.
\nTransposing a number (i.e., changing the side of the number) is the same as adding or subtracting the number from both sides of the equation. In doing so, we change the sign of the number.<\/p>\n
\nSolution:<\/strong>
\n
\nExample 2:<\/strong> A number increased by 15 gives 23. Find the number.
\nSolution:<\/strong> Let the number be x.
\nNumber increased by 15 = x + 15.
\nIt is 23.
\n\u2234 The equation is x + 15 = 23
\nor, \u00a0 \u00a0 \u00a0 \u00a0x + 15 – 15 = 23 – 15
\n(Subtracting 15 from both sides)
\nor, \u00a0 \u00a0 \u00a0 \u00a0x = 8
\nSo, the number = 8.<\/p>\n
\nSolution:<\/strong> Let the number be x.
\n\u2234 3 times of x = 3x
\n3x decreased by 6 = 3x – 6.
\nIt is 12.
\n\u2234 The equation is 3x – 6 = 12
\n3x – 6 + 6 = 12 + 6
\n(By adding 6 on both sides)
\nor, 3x = 18
\n3x\/3 = 18\/3
\n(Dividing by 3 on both sides)
\nor, x = 6
\nSo, the number = 6<\/p>\n","protected":false},"excerpt":{"rendered":"