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## G. Systems of Linear Equations

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**G. Systems of Linear Equations**Math 10: Foundations and Pre-Calculus**Key Terms:**• Find the definition of the following terms: • Systems of Linear Equations • Linear System • Solving by Substitution • Equivalent Systems • Solving by Elimination • Infinite • Coincident Lines**A school district has buses that carry 12 students and buses**that carry 24 students. The total capacity is 780. There are 20 more small buses than large buses. How many of each are there?**To determine how many of each bus there is we can write two**equations to model the situation. • First what are the unknown quantities. • Now lets use this info to set up an equation.**These 2 equations form a system of linear equations in 2**variables, s and l. Also referred to as a linear system. • A solution of a linear system is a pair of values of s and lthat satisfy both equations**Suppose you were told there were 35 small buses and 15 large**buses. To verify first see if it makes sense in the given data. • Our answers agree with the given data, so our selection is correct. • We can also verify our solutions by subbing the known values of s and l into the equations we made**For each equation, the left side is the same as the right**side so our answers are correct.**The example situation on p. 399 shows that it is better to**consider the given data to verify a solution first then check using the equations.**Practice**• Ex. 7.1 (p. 400) #1-14 #4-18**The solution of a linear system can be estimated by using**graphing. • If the 2 lines intersect, the coordinates (x,y) of the point of intersection are the solution to the system**The point of intersection appears to be (-2,-3)**• To verify the solution, check to see if the coordinate satisfies the equation**Practice**• Ex. 7.2 (p. 408) #1-16 odds in each #4-19 odds in each**3. The World of Technology**• Construct Understanding p. 411**Practice**• Ex. 7.3 (p. 412) #1-5 odds in each**Using graphing to solve systems of equations is time**consuming and we most times are only getting an approximate answer • We can use algebra to get an exact solution and one algebraic strategy is called solving by substitution**By using substitution, we transform a system of 2 linear**equations into a single equation in one variable. • Then we use what we know about solving linear equations to determine the value of the variable**These two linear systems have the same solution x=1 and y=2.**WHY?**They have the same solution because they are equivalent**equations in each system • Multiplying or dividing the equations in a linear system by a non-zero number does not change the graphs. • So their point of intersection, and hence, their solution is unchanged.**A system of equivalent equations is called an equivalent**linear system and has the same solution as the original**When an equation in a linear system has coefficients or a**constant term that are fractions, we can multiply by a common denominator to write an equivalent equation with integer coefficients • Basically we get ride of the fractions**Practice**• Ex. 7.4 (p. 424) #1-5, 7-14, 17-19, 21 #4-23 do the odds in the question that have multiply questions**5. Solving Systems of Equations using Elimination**• Construct Understanding p. 429**Similarly to the last section that we looked at, adding or**subtracting 2 equations in a linear system produces equivalent linear systems. • We use this property to solve a linear system by first eliminating one variable by adding or subtracting the two equations. • This is called Solving by Elimination**We may need to multiply one or both equations by a constant**before we can eliminate a variable by adding or subtracting**In a linear system, we may have to write equivalent**equations with integer coefficients before we apply the elimination strategy