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What is a linear system of equations?. A linear system of equations is two or more equations. The solution to a linear system is a point of intersection – or where the two equations intersect in the form (x, y). There are 4 methods for solving: Graphing Setting “y= y”
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What is a linear system of equations? A linear system of equations is two or more equations. The solution to a linear system is a point of intersection – or where the two equations intersect in the form (x, y). There are 4 methods for solving: Graphing Setting “y= y” Substitution (one variable in terms of the other) Elimination (also called Linear Combination) Set up Problem 4.1 A
How do I solve a linear system of equations by setting “y = y”? • Make sure both of the equations are written in slope intercept form (y=mx+b) • Set each equation equal to each other (leaving out the y) • Solve for x (variables will be on both sides of the equation) • Go back and solve for y by replacing what was found for “x” in either equation. Example: x + y = 10 x - y = 2 Step 1: Rewrite in y = mx + b y = -x +10 y = x - 2 Step 2: Set equations equal to each other -x + 10 = x -2 Step 3: Solve for x +x +x 10 = 2x - 2 +2 +2 12 = 2x 6 = x Step 4: Since x = 6 x + y = 10 6 + y =10 y = 4 So the solution is (6, 4)
How do I solve a linear system of equations by substitution? Make sure one of the equations are in slope-intercept form ** If not, rewrite one of them in slope-intercept form Rewrite the equation written in standard form by substituting in y into the equation Solve for x (you will need to use the distributive property) Use what you found for x to find the value of y Example #1: x + y =1 1st Yes, one the equations is in slope-intercept form y = x +7 x + (x+7) = 1 2ndRewrite the equation in standard form by substituting in x + 7 for y x + x + 7 = 1 3rd Solve for x 2x + 7 =1 -7 -7 2x = -6 x = -3 y = x +7 4th solve for y using x = -3 y= (-3) + 7 y = 4 So the solution is (-3, 4)
How do I solve a linear system of equations by elimination? Make sure equations are in standard form Look at the coefficients and decide whether x or y would be easiest to eliminate Decide what to multiply each equation by so that when you add the two, the variable will be eliminated. (Look for the least common multiple of the coefficients) Add the two equations together Solve for the remaining variable Example: 3x - y = 5 1st Are the equations in standard form? Yes 2x + 5y = -8 2nd Decide which variable would be easiest eliminate - ‘y’ since the coefficients are opposite in sign (5 and -1) 3rd multiply one of the equations so that the y variable will cancel out 5(3x - y = 5) -> 15x - 5y = 25 2x + 5y = -8 -> + 2x + 5y = -84th add the equations 17x = 17 x = 1 4th use x= 1 to find x by substituting into either equation: 3x – y = 5 3(1) - y = 5 3 - y = 5 -3 -3 -y = 2 -> y = -2 So the solution is (1, -2)
How do I solve a linear system of equations by substitution? Example #2: 2x + y = -1 x – 2y = 12 2x + y = -1 -> y = -2x -1 1st Rewrite in slope-intercept form x – 2y = 12 2nd Substitute in -2x -1 for y x – 2(-2x -1) = 12 3rd Solve for x (use the distributive property) x + 4x + 2 = 12 5x + 2 = 12 -2 -2 5x = 10 x = 2 y = -2x – 1 4th Solve for y y = -2(2) – 1 y = -4 – 1 y = -5 So the solution is (2, -5)
