lesson 16 solve systems of equations algebraically answer key
3 x+TT(T0P01P057S076Q(JUWSw5QpW w In all the systems of linear equations so far, the lines intersected and the solution was one point. x 4, { 2, { x A solution of a system of two linear equations is represented by an ordered pair (x, y). 2 Licensed under the Creative Commons Attribution 4.0 license. Each point on the line is a solution to the equation. y Solve the system of equations{3x+y=12x=y8{3x+y=12x=y8 by substitution and explain all your steps in words. There are infinitely many solutions to this system. If any coefficients are fractions, clear them. 3 Solve a System of Equations by Substitution We will use the same system we used first for graphing. Because the warm-up is intended to promote reasoning, discourage the useof graphing technology to graph the systems. { Step 3. + Invite students with different approaches to share later. The intersection of the given graphs is a point to the right of the vertical axis (and therefore having a positive \(x\)-value), so the graphs cannot represent that system. There will be times when we will want to know how many solutions there will be to a system of linear equations, but we might not actually have to find the solution. Some students may remember that the equation for such lines can be written as or , where and are constants. 2 + {2x3y=1212y+8x=48{2x3y=1212y+8x=48, Solve the system by substitution. y y In this case we will solve for the variable \(y\) in terms of \(x\): \[\begin{align*} The solution (if there is one)to thissystem would have to have-5 for the\(x\)-value. Activatingthis knowledge would enable students toquicklytell whether a system matches the given graphs. How many training sessions would make the salary options equal? If you're seeing this message, it means we're having trouble loading external resources on our website. 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\(4 + y = 12\): \(\begin {align} 4+y&=12\\4 + (2x-7) &=12\\4 + 2x - 7 &=12\\ 2x -7 + 4 &=12\\ 2x-3&=12\\2x &=15\\x &=7.5\\ \\y&=2x - 7\\y&=2(7.5) - 7\\ y&=15-7\\y&=8 \end{align}\). y We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. \(\begin{cases}{y=2x+1} \\ {y=4x1}\end{cases}\), Solve the system by graphing: \(\begin{cases}{y=2x+2} \\ {y=-x4}\end{cases}\), Solve the system by graphing: \(\begin{cases}{y=3x+3} \\ {y=-x+7}\end{cases}\). Inexplaining their strategies, students need to be precise in their word choice and use of language (MP6). 4 7 Find the numbers. y5 3x2 2 y5x1 1 Prerequisite: Find the Number of Solutions of a System Study the example showing a system of linear equations with no solution. Step 3: Solve for the remaining variable. 5 x+70-10 x &=40 \quad \text{distribute 10 into the parentheses} \\ If you missed this problem, review Example 2.34. Rewriting the originalequationthis way allows us to isolatethe variable \(q\). = & y = 3x-1 & y=3x-6 \\ &m = 3 & m = 3 \\&b=-1 &b=-6 \\ \text{Since the slopes are the same andy-intercepts} \\ \text{are different, the lines are parallel.}\end{array}\). Solve the system by substitution. = Now that we know how to solve systems by substitution, thats what well do in Step 5. How many cars would need to be sold to make the total pay the same? The first company pays a salary of $10,000 plus a commission of $1,000 for each car sold. 1 { & 3 x+8 y=78 \\ If we subtract \(3p\) from each side of the first equation,\(3p + q = 71\), we get an equivalent equation:\(q= 71 - 3p\). 5 stream y x x \end{align*}\nonumber\]. Since every point on the line makes both equations. 2 4 + = 3 Ask these students to share later. x y We call a system of equations like this an inconsistent system. 4 { OpenStax is part of Rice University, which is a 501(c)(3) nonprofit. 3, { A system with parallel lines, like Exercise \(\PageIndex{19}\), has no solution. = 10 = y }{=}4 \cdot 1-1} \\ {3=3 \checkmark}&{3=3 \checkmark} \end{array}\), \(\begin{aligned} 3 x+y &=-1 \\ y &=-3 x-1 \\ m &=-3 \\ b &=-1 \\ 2 x+y &=0 \\ y &=-2 x \\ b &=0 \end{aligned}\), \(\begin{array}{rllrll}{3x+y}&{=}&{-1} & {2x +y}&{=}&{0}\\{3(-1)+ 2}&{\stackrel{? 2 Using the distributive property, we rewrite the first equation as: Now we are ready to add the two equations to eliminate the variable \(x\) and solve the resulting equation for \(y\) : \[\begin{array}{llll} y 6 8 Lesson 13 Solving Systems of Equations; Lesson 14 Solving More Systems; Lesson 15 Writing Systems of Equations; Let's Put It to Work. y By the end of this section, you will be able to: Before you get started, take this readiness quiz. + Substitute the solution in Step 3 into one of the original equations to find the other variable. y y x + Name what we are looking for. Solve the system by substitution. The solution of the linear system of equations is the intersection of the two equations. 3 A system of equations whose graphs are intersect has 1 solution and is consistent and independent. The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics. 6. y x 2 How many quarts of fruit juice and how many quarts of club soda does Sondra need? y 1 (3)(-3 x & + & 2 y & = & (3) 3 \\ y Line 2 is exactly vertical and intersects around the middle of Line 1.
. Manny needs 3 quarts juice concentrate and 9 quarts water. 8 The second equation is already solved for y. This page titled 5.1: Solve Systems of Equations by Graphing is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. }{=}2 \cdot 1+1} &{3\stackrel{? = y 3 x+8 y=78 3a+4b=9 -3a-2b=-3. apps. First, solve the first equation \(6 x+2 y=72\) for \(y:\), \[\begin{array}{rrr} = y 15 3 However, there are many cases where solving a system by graphing is inconvenient or imprecise. x USE A PROBLEM SOLVING STRATEGY FOR SYSTEMS OF LINEAR EQUATIONS. Monitor for the different ways that students use substitutions to solve the systems. y Choosing the variable names is easier when all you need to do is write down two letters. We are looking for the number of quarts of fruit juice and the number of quarts of club soda that Sondra will need. = y x & + &y & = & 7 \\ Substitute the expression from Step 1 into the other equation. create. Columbus, OH: McGraw-Hill Education, 2014. {4x+2y=46xy=8{4x+2y=46xy=8. 15, { 4 \(\begin{cases}{3x+y=1} \\ {2x+y=0}\end{cases}\), Solve each system by graphing: \(\begin{cases}{x+y=1} \\ {2x+y=10}\end{cases}\), Solve each system by graphing: \(\begin{cases}{ 2x+y=6} \\ {x+y=1}\end{cases}\). 4 8 x { y If you write the second equation in Exercise \(\PageIndex{22}\) in slope-intercept form, you may recognize that the equations have the same slope and same y-intercept. Rearranging or solving \(4+ y=12\) to get \(y =8\), and then substituting 8 for \(y\) in the equation\(y=2x - 7\): \(\begin {align} y&=2x - 7\\8&=2x - 7\\ 15&=2x \\ 7.5 &=x\end{align}\). 3 + 7 + Answer: (1, 2) Sometimes linear systems are not given in standard form. x For access, consult one of our IM Certified Partners. y How many ounces of coffee and how many ounces of milk does Alisha need? Determine if each of these systems could be represented by the graphs. 1 y 2 y 1, { 5 = x y x 3 + (4, 3) does not make both equations true. 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. 7 3 In Example 5.16 it will be easier to solve for x. \(\begin{cases} x + 2y = 8 \\x = \text-5 \end{cases}\), \(\begin{cases} y = \text-7x + 13 \\y = \text-1 \end{cases}\), \(\begin{cases} 3x = 8\\3x + y = 15 \end{cases}\), \(\begin{cases} y = 2x - 7\\4 + y = 12 \end{cases}\). 44 0, { 8 5 In Example 27.2 we will see a system with no solution. Spelling questions (x) are worth 5 points and vocabulary questions (y) are worth 10 points. 7. y Solve the system by graphing: \(\begin{cases}{y=\frac{1}{2}x3} \\ {x2y=4}\end{cases}\), Solve each system by graphing: \(\begin{cases}{y=-\frac{1}{4}x+2} \\ {x+4y=-8}\end{cases}\), Solve each system by graphing: \(\begin{cases}{y=3x1} \\ {6x2y=6}\end{cases}\), Solve the system by graphing: \(\begin{cases}{y=2x3} \\ {6x+3y=9}\end{cases}\), Solve each system by graphing: \(\begin{cases}{y=3x6} \\ {6x+2y=12}\end{cases}\), Solve each system by graphing: \(\begin{cases}{y=\frac{1}{2}x4} \\ {2x4y=16}\end{cases}\). We will solve the first equation for xx and then substitute the expression into the second equation. 8, { 3 The solution of a system of equations are the values of its variables which, when substituted into the two original equations, give us true statements. As students work, pay attention to the methods students use to solve the systems. x 3 If some students struggle with the last system because the variable that is already isolated is equal to an expression rather than a number, askwhat they would do if the first equation were \(y= \text{a number}\)instead of \(y=2x-7\). 2 y {x4y=43x+4y=0{x4y=43x+4y=0, Solve the system by substitution. 8 0 obj 3 + 1 2 2 x 142 L16: Solve Systems of Equations Algebraically Read the problem below. Systems of equations with graphing Get 3 of 4 questions to level up! { The first company pays a salary of $12,000 plus a commission of $100 for each policy sold. \[\left(\begin{array}{l} 3 Solve the system by substitution. y \end{array}\right)\nonumber\], Again, here we solve the system of equations using substitution. x \end{array}\right)\nonumber\], \[-1 x=-3 \quad \Longrightarrow \quad x=3\nonumber\], To find \(y,\) we can substitute \(x=3\) into the first equation (or the second equation) of the original system to solve for \(y:\), \[-3(3)+2 y=3 \Longrightarrow-9+2 y=3 \Longrightarrow 2 y=12 \Longrightarrow y=6\nonumber\]. x x { "5.1E:_Exercises" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.Montana Diesel Strain,
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