UNIT 5. SIMULTANEOUS EQUATIONS

3º ESO. Definitions UNIT 5. SIMULTANEOUS EQUATIONS A linear equation with two unknowns is an equation with two unknowns having both of them degree one. Eamples. 3 + 5 and + 6 9. The standard form for these equations is: a + b c the variables go in order on the left side and the constant term is on the right. Simultaneous linear equations are a set of linear equations made up of several linear equations considered at the same time. Another name is sstem of linear equations. Eample. + 4 This is the standard form for linear equations when the are part of a sstem of equations. The brace, }, on the right indicates that the two equations are intended to be solved simultaneousl. Eample. + 0 If we onl consider the equation + 0, we have two unknowns and man pairs of values of and fit the equation: (Infinitel man solutions). 9 5 + 0 0 0 0 0 is another equation with the same two unknowns. Again, (infinitel) man pairs of values of and fit the equation: 4 4.9 3 6 There is onl one pair of values of and which fit both equations: 6 and 4, because 6 + 4 0 and 6 4.

3º ESO Definition. For a pair of simultaneous equations, a solution is a pair (,) of numbers that make both equations true when substituted into the equations. To solve simultaneous equations ou need to find values which fit both equations simultaneousl. Remark: the solutions can be negative integers and also fractions. Sometimes it is useful to label the equations with capital letters as Equation or () and Equation or ().. Equivalent equations We sa that two equations are equivalent if the have the same solutions. For eample, equation 3 6 is equivalent to 4 8, their solutions are both. Two sstems are equivalent if their solutions are the same. Rule. If ou do the same to both members of an equation, the equalit is still true and the new equation is equivalent to the original one. This means: Was to get equivalent equations You can add/subtract the same number to both members. (Transpose terms) For eample: + 9, subtracting 9 from both members, we get 8. 5 6 0 is equivalent to 5 6, adding 0. You can multipl/divide all the terms b the same number. For eample: 4 8 is equivalent to 4, dividing b. If an equation is multiplied b (), all the coefficients Eample: " 4, then 4 change their sign. For a pair of simultaneous equations, one of the equations can be substituted b an equivalent one. The simultaneous equations obtained are said to be equivalent to the original ones: the solution is still the same. And this will be the wa to solve them. Eercise. Give equivalent simultaneous equations to the following pairs, so that the are written in standard form or in a simpler form. a) + 0 0 4 + b) 4 0 5 5 c) 9 + 6 39

3º ESO 3 3. The graph method Later on, we will stud three algebraic methods for solving simultaneous equations. But firts we are going to stud the graph method. You could be asked to do all the methods in the eam so make sure ou learn them all. Vocabular. to plot, sketch: representar gráficamente, trazar. Similar to graph. Take into account: ) When we graph a linear equation in two variables as a line in the plane, all the points on this line correspond to ordered pairs of numbers that satisf the equation. To graph a function we need the epression of, in terms of, that is: f(). Eamples: Sketch the following lines on the same cartesian sstem. Use different colors. How?. Do a table to find several points.. Join the points. 3. Draw a line. Y 3-3 - - 3 X - - -3 Eample : Plot the function + 3. Remember two concepts: Slope (pendiente). (As it is positive, the line rises to the right) -intercept (ordenada en el origen) 3. Eample : Plot the function 3 Slope. -intercept Eample 3: Plot the function + Slope (As it is negative, the line rises to the left) -intercept

3º ESO 4 ) When we graph two linear equations, all the points of intersection the points which lie on both lines are the points which satisf both equations. 3) To solve a sstem of equations b graphing, graph all the equations in the sstem. The point(s) at which all the lines intersect are the solutions to the sstem. METHOD 0. Work out the s.. Do a table of at least two values for both equations. This gives as points.. Draw the two graphs. 3. Find the X- and Y-values where the cross. Simple rule for the graph method: THE SOLUTION of two SIMULTANEOUS EQUATIONS is simpl the X and Y WHERE THEIR GRAPHS CROSS. Eample. Solve + 4 Step 0. Work out the s so that we can see these equations as functions.! " # + 4 $ % + + 4 Step. Do a table for each equation. Choose the, calculate the Table for + 0 3 5 Table for +4 0 4 5 4 0 Step. Draw the graphs: Step 3. The solution is, 3, because the lines cross at the point (, 3).

3º ESO 5 Eample. Solve the following sstem of linear equations: + 5 Solution: Step 0. To represent the two lines, we need to epress the equations as functions. We need to work out the s, even if it takes us some steps: 5 + 5 5 Step. To find some points of the graph, we calculate some coordinates. Table for 5 0 5 Table for 3 Step. Draw the graphs: Step 3. The solution is 3,, because the lines cross at the point (3, ). Solution

3º ESO IES Miguel Espinosa 4. Classification of simultaneous equations The simultaneous equations can be classified attending to their solutions. Tpe : Consistent equations. A. Consistent independent equations: One unique solution. The graph method gives us two intersecting lines that cross at one point. Eample. + 0 4 has one unique solution: (7, 3) B: Consistent dependent equations: Infinitel man solutions. The graph method gives us two coincident lines. + 5 Eample. has infinite solutions, like (, 3), (, 4), + 0 (.5, 3.5), (9, 4), etc. This happens when the equations are equivalent, so one of them does not give an information. Tpe : Inconsistent equations: No solutions. The graph method gives us two parallel lines (The never cross). Eample. + has not an solution. + 4 Both equations are incompatible with each other. Author: Susana López 6

3º ESO IES Miguel Espinosa 5. Algebraic methods for solving simultaneous equations Simultaneous equations can be solved using different methods. All methods are based in rules for getting equivalent equations. Notice that steps 3, 4 and 5 are alwas the same. Several was to solve simultaneous equations means several was to have fun. Think about it Jo of jos!! 5.. The substitution method (Sustitución). Work out one of the unknowns from one equation.. Substitute the epression into the other equation. 3. Work out this equation with onl one unknown. 4. Get the value of the other unknown b substituting into one of the original equations. 5. Check the solution b substituting. ( A) + Eample. ( B) 4. We choose and equation (A). Work out :. Substitute into equation (B): ( ) 4 ONLY ONE UNKNOWN IN THIS STEP 3. Work out the from the equation we have got 4 4 8 4 4. Get the value of : 4 8 8 5. Check the solution b substituting into the original equations: (A) 8 + 4 (B) 8 4 4 Author: Susana López 7

3º ESO IES Miguel Espinosa 5.. Equating unknowns method.. Work out one of the unknowns from both equations.. Equate both epressions of the same unknown. (to equate: igualar) 3. Work out this equation with onl one unknown. 4. Get the value of the other unknown. 5. Check the solution b substituting. Eample. ( A ) 3 ( B) 3 + 3 From (A) we get: 3 +. From (B) we have:. We choose to be worked out from both equations. 3. 3. Now we equate both epressions 3 + 3 of : 3 ONLY ONE UNKNOWN IS LEFT IN THIS STEP 3. Solve this new equation, where there is not an : 3 + 3 3 3( 3 + ) (3 ) 9 + 3 6 4 3 + 4 6 + 9 7 35 This gives the solution 5 4. Get the value of substituting the value of into one of the epressions we have got in step : 3 + 3 + 5 5. Check the solution: (A) "# 5 #3 $ % (B) 3"+ " 5 3& Author: Susana López 8

3º ESO IES Miguel Espinosa 5. 3. The elimination method or The addition or subtraction method Both equations must be in the standard form a + b c. (constant at the right) So ou ma have to rearrange the equations ou are given. Adding and subtracting equations is an allowed (and needed) operation to get equivalent sstems.. Get the same coefficient for the same unknown. Add or subtract both equations. One unknown will be eliminated. 3. Solve the remaining equation with one unknown. 4. Get the value of the other unknown. 5. Check the solution b substituting. Remember: In step : In general the new coefficient will be the l.c.m of the coefficients of the chosen unknown. In step : To eliminate terms with opposite signs, add the equations. To eliminate terms with the same signs, subtract the equations. Attention to the signs! ( A) 3 Eample. ( B) +. Get the same coefficient for the same unknown. We choose, because it has alread different sign in (A) and (B). l.c.m. (, 3) 6. So we do: (A) and (B) 3 4! 6 " # 3 + 6!3$ (These simultaneous equations are equivalent to A and B together). Add both equations: 4! 6 4" # 3 + 6!3$ % ONLY ONE UNKNOWN LEFT 7 3. Solve the equation: 7 3 4. Get the value of b substituting into equation (B) the value of : 3 4 (B) + 5. Check the solution: (A): 3 3 ( ) (B): 3 ( ) + Author: Susana López 9

3º ESO Eercises Eercise. Use the three algebraic methods studied to solve each of the following simultaneous equations. Check the solutions b substituting (onl once in each case). a) + 8 b) + 0 8 c) 5 + d) + 8 + 7 e) 3 + 4 4 + Solutions. a) (6, ). b) (,4) 6. c), 3 3. d) (,3). e) (4, 3). Eercise. Use the graphic method to solve each of these pairs of simultaneous equations. + 6 + 8 + 8 a) b) c) + 7 3 + + 3 6 3 + 4 4 d) e) f) + 5 + Solutions. a) (4, ). b) (3, 5). c) (,3). d) (,3). e) (3, ). f) (4, 3). Word Problems. Pam bus 6 pencils and 3 pens for 93 pence. Ra bus pencils and 5 pens for 9 pence. Write down two equations connecting the prices of pencils and pens. Find the cost of a pencil and the cost of a pen. Solution: 8p, 5p.. 0 dozen standard eggs and 5 dozen small eggs cost 3.60. 5 dozen standard eggs and 8 dozen small eggs cost.3. Find the cost of a dozen of standard eggs and the cost of a dozen of small eggs. Solution: 95p, 8p. 3. Jenn tpes at words per minute. Stuart tpes at words per minute. When Jenn and Stuart both tpe for minute the tpe a total of 70 words. When Jenn tpes for 5 minutes and Stuart tpes for 3 minutes the tpe a total of 70 words. Calculate and. Sol: 00 words and 70 words per minute. 4. At a café, John bus 3 coffees and teas for.30 and Susan bus coffees and 3 teas for.0. Calculate the price of a coffee and the price of a tea. Solution: 50p, 40p. 5. Charlotte and Brendan are buing presents. Charlotte bus one perfume spra and one bottle of aftershave for 5. Brendan bus three perfume spras and two bottles of aftershave for.find the cost of a perfume spra and the cost of a bottle of aftershave. Sol: perfume spra and aftershave 3. Susana López 0

3º ESO 6. Kate has found a puzzle in a newspaper. Here is Kate s newspaper problem. 0 6 She sees that four squares add up to 6. Two triangles and two circles add up to 0. Two triangles, one circle and one square add up to. She has to find the value of each shape. Solution: triangle 3, circle,square 4. 7. The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased b 7. Find the number. Solution: 5. Of our interest: http://www.saab.org/mathdrills/lineq.cgi http://regentsprep.org/regents/math/math-a.cfm http://regentsprep.org/regents/math/sslin/algssadd.htm Susana López