y {5x2y=10y=52x{5x2y=10y=52x. x The coefficients of the \(x\) variable in our two equations are 1 and \(5 .\) We can make the coefficients of \(x\) to be additive inverses by multiplying the first equation by \(-5\) and keeping the second equation untouched: \[\left(\begin{array}{lllll} In all the systems of linear equations so far, the lines intersected and the solution was one point. In the following exercises, solve the systems of equations by substitution. y Substitution method for systems of equations. Exercise 5 . \(\begin{cases}{ f+c=10} \\ {f=4c}\end{cases}\). {x6y=62x4y=4{x6y=62x4y=4. + 2 + 2 {x2y=23x+2y=34{x2y=23x+2y=34. y 4, { { \end{array}\nonumber\], To find \(x,\) we can substitute \(y=1\) into either equation of the original system to solve for \(x:\), \[x+1=7 \quad \Longrightarrow \quad x=6\nonumber\]. Make sure you sign-in x { This should result in a linear equation with only one variable. Solve for yy: 8y8=322y8y8=322y = y If the equations are given in standard form, well need to start by solving for one of the variables. Invite students with different approaches to share later. {2x+y=11x+3y=9{2x+y=11x+3y=9, Solve the system by substitution. Hence, we get \(x=6 .\) To find \(y,\) we substitute \(x=6\) into the first equation of the system and solve for \(y\) (Note: We may substitute \(x=6\) into either of the two original equations or the equation \(y=7-x\) ): \[\begin{array}{l} Usually when equations are given in standard form, the most convenient way to graph them is by using the intercepts. 2 {x+y=44xy=2{x+y=44xy=2. We will use the same problem solving strategy we used in Math Models to set up and solve applications of systems of linear equations. {5x+2y=124y10x=24{5x+2y=124y10x=24. 0, { (2)(4 x & - & 3 y & = & (2)(-6) << /ProcSet [ /PDF ] /XObject << /Fm4 19 0 R >> >> The solution (if there is one)to thissystem would have to have-5 for the\(x\)-value. USE A PROBLEM SOLVING STRATEGY FOR SYSTEMS OF LINEAR EQUATIONS. Exercise 2. 3 y 2 Step 5. { In this chapter we will use three methods to solve a system of linear equations. The sum of two numbers is 15. + To illustrate, we will solve the system above with this method. 2 A student has some $1 bills and $5 bills in his wallet. Solve the system by substitution. We will first solve one of the equations for either x or y. 1 Exercise 3. 3 Step 6. 30 = (2, 1) does not make both equations true. If the graphs extend beyond the small grid with x and y both between 10 and 10, graphing the lines may be cumbersome. 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Algebraically, [ "article:topic", "substitution method", "showtoc:no", "license:ccbyncnd", "elimination method", "authorname:elhittietal", "licenseversion:40" ], https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FBookshelves%2FAlgebra%2FBook%253A_Arithmetic_and_Algebra_(ElHitti_Bonanome_Carley_Tradler_and_Zhou)%2F01%253A_Chapters%2F1.29%253A_Solving_a_System_of_Equations_Algebraically, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), 1.30: Solving a System of Equations Graphically, Samar ElHitti, Marianna Bonanome, Holly Carley, Thomas Tradler, & Lin Zhou, CUNY New York City College of Technology & NYC College of Technology, New York City College of Technology at CUNY Academic Works, ElHitti, Bonanome, Carley, Tradler, & Zhou. 9 Finally, we check our solution and make sure it makes both equations true. = The measure of one of the small angles of a right triangle is 26 more than 3 times the measure of the other small angle. x 5 Solve by elimination: {5x + 12y = 11 3y = 4x + 1. This leaves you with an equivalent equation with one variable, which can be solved using the techniques learned up to this point. Hence \(x=10 .\) Now substituting \(x=10\) into the equation \(y=-3 x+36\) yields \(y=6,\) so the solution to the system of equations is \(x=10, y=6 .\) The final step is left for the reader. 7, { = + consent of Rice University. y 3 The second pays a salary of $20,000 plus a commission of $500 for each car sold. x then you must include on every digital page view the following attribution: Use the information below to generate a citation. 5 8 y 2 = Solve the system of equations{x+y=10xy=6{x+y=10xy=6. 8 = TO SOLVE A SYSTEM OF LINEAR EQUATIONS BY GRAPHING. -5 x+70 &=40 \quad \text{collect like terms} \\ Accessibility StatementFor more information contact us atinfo@libretexts.org. x = y 8 y Solve the system by substitution. See the image attribution section for more information. Simplify 42(n+5)42(n+5). 1 y \Longrightarrow & x=10 Remind students that if \(p\) is equal to \(2m+10\), then \(2p\)is 2 times \(2m+10\) or \(2(2m+10)\). The measure of one of the small angles of a right triangle is 18 less than twice the measure of the other small angle. 1 2 11. Check to make sure it is a solution to both equations. { \end{array}\nonumber\], Therefore the solution to the system of linear equations is. The first company pays a salary of $ 14,000 plus a commission of $100 for each cable package sold. Consider collecting students' responses or asking them to share their written arguments with a partner. x & + &y & = & 7 \\ 44 A system of equations that has at least one solution is called a consistent system. Multiply one or both equations by a nonzero number so that the coefficients of one of the variables are additive inverses. Without graphing, determine the number of solutions and then classify the system of equations. Solve the linear equation for the remaining variable. 4 y = {2xy=1y=3x6{2xy=1y=3x6. The measure of one of the small angles of a right triangle is 45 less than twice the measure of the other small angle. y used to solve a system of equations by adding terms vertically this will cause one of the variables to be . + x 2 How many quarts of fruit juice and how many quarts of club soda does Sondra need? Find the numbers. y Check the answer in the problem and make sure it makes sense. Substitute the expression found in step 1 into the other equation. 5 Its graph is a line. \end{align*}\nonumber\]. 4 2 In this case we will solve for the variable \(y\) in terms of \(x\): \[\begin{align*} + Solve the system. The length is 5 more than three times the width. 142 L16: Solve Systems of Equations Algebraically Read the problem below. Practice Solving systems with substitution Learn Systems of equations with substitution: 2y=x+7 & x=y-4 Systems of equations with substitution Systems of equations with substitution: y=4x-17.5 & y+2x=6.5 Systems of equations with substitution: -3x-4y=-2 & y=2x-5 4 Coincident lines have the same slope and same y-intercept. << /ProcSet [ /PDF ] /XObject << /Fm3 15 0 R >> >> {3x+y=52x+4y=10{3x+y=52x+4y=10. = x into \(3x+8=15\): \(\begin {align} 3x&=8\\x&=\frac83\\ \\3x+y &=15\\ 3(\frac83) + y &=15\\8+y &=15\\y&=7 \end{align}\). + Solve a System of Equations by Substitution We will use the same system we used first for graphing. = 4 2 7 Choosing the variable names is easier when all you need to do is write down two letters. Chapter 1 - The Language Of Algebra Chapter 1.1 - A Plan For Problem Solving Chapter 1.2 - Words And Expressions Chapter 1.3 - Variables And Expressions Chapter 1.4 - Properties Of Numbers Chapter 1.5 - Problem-solving Strategies Chapter 1.6 - Ordered Pairs And Relations Chapter 1.7 - Words, Equations, Tables, And Graphs Chapter 2 - Operations = Systems of equations with graphing Get 3 of 4 questions to level up! + { 3 endstream y 7. + The first company pays a salary of $10,000 plus a commission of $1,000 for each car sold. x endobj {y=x+10y=14x{y=x+10y=14x. Since every point on the line makes both equations. Does a rectangle with length 31 and width. A system of equations whose graphs are parallel lines has no solution and is inconsistent and independent. 10 The basic idea of the method is to get the coefficients of one of the variables in the two equations to be additive inverses, such as -3 and \(3,\) so that after the two equations are added, this variable is eliminated. Find the measure of both angles. 5 y Display their work for all to see. 3 Since 0 = 10 is a false statement the equations are inconsistent. = + 7x+2y=-8 8y=4x. When we graph two dependent equations, we get coincident lines. In each of these two systems, students are likely to notice that one way of substituting is much quicker than the other. = 5 When two or more linear equations are grouped together, they form a system of linear equations. 6 0 obj We are looking for the number of training sessions. 7 0 obj Licensed under the Creative Commons Attribution 4.0 license. + The equation above can now be solved for \(x\) since it only involves one variable: \[\begin{align*} Coincident lines have the same slope and same y-intercept. 0 12 In the Example 5.22, well use the formula for the perimeter of a rectangle, P = 2L + 2W. y y We can choose either equation and solve for either variablebut well try to make a choice that will keep the work easy. The equations are dependent. Make sure you sign-in Solve the system by graphing: \(\begin{cases}{3x+y=1} \\ {2x+y=0}\end{cases}\), Well solve both of these equations for yy so that we can easily graph them using their slopes and y-intercepts. 4 11 }{=}}&{6} &{2(-3) + 3(6)}&{\stackrel{? Substituting the value of \(3x\) into \(3x+8=15\): \(\begin {align} 3x+y &=15\\ 8 + y &=15\\y&=7 \end{align}\). 5 stream + A system of equations whose graphs are intersect has 1 solution and is consistent and independent. 2 Which method do you prefer? 2 Sometimes, we need to multiply both equations by two different numbers to make the coefficients of one of the variables additive inverses. An example of a system of two linear equations is shown below. y x Lets take one more look at our equations in Exercise \(\PageIndex{19}\) that gave us parallel lines. 7 = = = are not subject to the Creative Commons license and may not be reproduced without the prior and express written 2 Substitute \(y=-3 x+36\) into the second equation \(3 x+8 y=78\) : \[\begin{align*} The length is five more than twice the width. endobj We will use the same system we used first for graphing. y y Then we substitute that expression into the other equation. Ask students to share their strategies for each problem. In the last system, a simple rearrangement to one equation would put it inthis form.) = 1 Give students 68minutes of quiet time to solve as many systems as they can and then a couple of minutes to share their responses and strategies with their partner. Instructional Video-Solve Linear Systems by Substitution, Instructional Video-Solve by Substitution, https://openstax.org/books/elementary-algebra-2e/pages/1-introduction, https://openstax.org/books/elementary-algebra-2e/pages/5-2-solving-systems-of-equations-by-substitution, Creative Commons Attribution 4.0 International License, The second equation is already solved for. >o|o0]^kTt^ /n_z-6tmOM_|M^}xnpwKQ_7O|C~5?^YOh = 4, { Solutions of a system of equations are the values of the variables that make all the equations true. 3 x+TT(T0 B3C#sK#Tp}\#|@ x = y x y x = 3 stream y y 2 x 2 2y 5 4 3y 5 2 0.5 x 1 2 Model It You can use elimination to solve for one variable. = = x { x << /Length 8 0 R /Filter /FlateDecode /Type /XObject /Subtype /Form /FormType Solve systems of linear equations by using the linear combinations method, Solve pairs of linear equations using patterns, Solve linear systems algebraically using substitution. We need to solve one equation for one variable. 8 }{=}}&{-1} &{2(-1)+2}&{\stackrel{? 2 { + { If you missed this problem, review Example 2.34. x + 2 2 Print.7-3/Course 2: Book Pages and Examples The McGraw-Hill Companies, Inc. Glencoe Math Course 2 The perimeter of a rectangle is 58. x (Alternatively, use an example with a sum of two numbers for\(p\): Suppose \(p=10\), which means \(2p=2(10)\) or 20. The first method we'll use is graphing. They are parallel lines. 5 Manny needs 3 quarts juice concentrate and 9 quarts water. Solve the system by substitution. }{=}}&{0} \\ {-1}&{=}&{-1 \checkmark}&{0}&{=}&{0 \checkmark} \end{array}\), \(\begin{aligned} x+y &=2 \quad x+y=2 \\ 0+y &=2 \quad x+0=2 \\ y &=2 \quad x=2 \end{aligned}\), \begin{array}{rlr}{x-y} & {=4} &{x-y} &{= 4} \\ {0-y} & {=4} & {x-0} & {=4} \\{-y} & {=4} & {x}&{=4}\\ {y} & {=-4}\end{array}, We know the first equation represents a horizontal, The second equation is most conveniently graphed, \(\begin{array}{rllrll}{y}&{=}&{6} & {2x+3y}&{=}&{12}\\{6}&{\stackrel{? = y=-x+2 Unit: Unit 4: Linear equations and linear systems, Intro to equations with variables on both sides, Equations with variables on both sides: 20-7x=6x-6, Equations with variables on both sides: decimals & fractions, Equations with parentheses: decimals & fractions, Equation practice with complementary angles, Equation practice with supplementary angles, Creating an equation with infinitely many solutions, Number of solutions to equations challenge, Worked example: number of solutions to equations, Level up on the above skills and collect up to 800 Mastery points, Systems of equations: trolls, tolls (1 of 2), Systems of equations: trolls, tolls (2 of 2), Systems of equations with graphing: y=7/5x-5 & y=3/5x-1, Number of solutions to a system of equations graphically, Systems of equations with substitution: y=-1/4x+100 & y=-1/4x+120, Number of solutions to a system of equations algebraically, Number of solutions to system of equations review, Systems of equations with substitution: 2y=x+7 & x=y-4, Systems of equations with substitution: y=4x-17.5 & y+2x=6.5, Systems of equations with substitution: y=-5x+8 & 10x+2y=-2, Substitution method review (systems of equations), Level up on the above skills and collect up to 400 Mastery points, System of equations word problem: no solution, Systems of equations with substitution: coins. y x 4 6 Here are two ways of solving the last system,\(\begin{cases} y = 2x - 7\\4 + y = 12 \end{cases}\),by substitution: Substituting \(2x - 7\) for \(y\) in the equation\(4 + y = 12\): \(\begin {align} 4+y&=12\\4 + (2x-7) &=12\\4 + 2x - 7 &=12\\ 2x -7 + 4 &=12\\ 2x-3&=12\\2x &=15\\x &=7.5\\ \\y&=2x - 7\\y&=2(7.5) - 7\\ y&=15-7\\y&=8 \end{align}\). x 2 y The number of quarts of fruit juice is 4 times the number of quarts of club soda. The systems of equations in Exercise \(\PageIndex{4}\) through Exercise \(\PageIndex{16}\) all had two intersecting lines. 6 \Longrightarrow & y=7-x Description: Graph of 2 intersecting lines, origin O, in first quadrant. It must be checked that \(x=10\) and \(y=6\) give true statements when substituted into the original system of equations. y A system of two linear equations in two variables may have one solution, no solutions, or infinitely many solutions. = y 3 2 The second equation is already solved for y, so we can substitute for y in the first equation. 4 y Solve this system of equations. 2 y Here are two ways for solving the third system,\(\begin{cases} 3x = 8\\3x + y = 15 \end{cases} \), by substitution: Findingthe value of \(x\) and substituting it citation tool such as, Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis. = Follow with a whole-class discussion. Section Lesson 16: Solve Systems of Equations Algebraically Section Lesson 17: Performance Task Page 123: Prerequisite: Identify Proportional Relationships Page 125: Use Tables, Graphs and Equations Page 127: Compare Proportional Relationships Page 129: Represent Proportional Relationships Exercise 1 Exercise 2 Exercise 3 Exercise 4 Exercise 5 8 = As students work, pay attention to the methods students use to solve the systems. x 1 Since we get the false statement \(2=1,\) the system of equations has no solution. The perimeter of a rectangle is 58. \end{array}\right) \Longrightarrow\left(\begin{array}{lllll} We will solve the first equation for x. Maxim has been offered positions by two car dealers. Solve a system of equations by substitution. Some students who correctly write \(2m-2(2m+10)=\text-6\) may fail to distribute the subtraction and write the left side as\(2m-4m+20\).
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