Systems of Linear Equations and Inequalities

The length is one more than twice the width. P = 26 inches The length is one more than twice the width. Let x = width, y = length 2x + 2y = 26 y = 2x + 1 (4, 9)

Solving a System of Linear Equations, Method 1 Substitution Step 1: Solve one equation for x (or for y). Step 2: Substitute the value you found in Step 1 into the other equation. The result is an equation with one variable. Step 3: Solve this new equation for y (or for x). Step 4: Use the value you found in Step 3 for y (or for x) and substitute it into one of the original equations. Solve for x (or for y). Given the system: x + 3y = 12 2x + y = 9 Use substitution to write the solution as an ordered pair (x, y).

Solving a System of Linear Equations, Method 2 Elimination Step 1: Choose one variable to eliminate. Step 2: Choose a multiplier for one or both equations, so that the equations have opposite terms. Multiply by the multiplier(s). Step 3: Add the two equations. One variable will be eliminated. Step 4: Solve for the other variable. Use elimination to write the solution as an ordered pair (x, y). Given the system: x + 3y = 12 2x + y = 9 Step 5: Replace the solved variable with its value. Solve for the remaining variable.

Solving a System of Linear Equations, Method 3 Graphing Step 2: Graph both equations on the same coordinate grid. Step 3: Locate the point(s) where the two lines intersect. Use graphing to find the solution. Given the system: x + 3y = 12 2x + y = 9 Step 1: Write both equations in slope-intercept form.

Classifying Systems of Equations consistent — has at least one solution dependent — has infinite solutions independent — has exactly one solution inconsistent — has no solution Classify the systems as dependent consistent, independent consistent, or inconsistent 3x + y = 9 6x + 2y = 18 5 – 2y = x x + 3y = 6 y = 3x + 7 y – 1 = 3x

Classifying Systems of Equations consistent — has at least one solution dependent — has infinite solutions independent — has exactly one solution inconsistent — has no solution Classify the systems as dependent consistent, independent consistent, or inconsistent 3x + y = 9 6x + 2y = 18 5 – 2y = x x + 3y = 6 y = 3x + 7 y – 1 = 3x dependent consistent independent consistent inconsistent

Classifying Graphs of Systems of Equations dependent consistent independent consistent inconsistent infinite one none same different same or different Number of Solutions Slopes y-intercepts Classification Type of Graph

Solving a System of Linear Inequalities Step 1: Choose one variable to eliminate. Step 2: Choose a multiplier for one or both equations, so that the Step 3: Add the two equations. One variable will be eliminated. Step 4: Solve for the other variable. Graph to find the solution region. Given the system: x + 3y ≥ 12 2x + y < 9 Step 5: Replace the solved variable with its value. equations have opposite terms. Multiply by the multipliers. Solve for the remaining variable.

Solving a System of Linear Inequalities Graph to find the solution region. Given the system: x + 3y ≥ 12 2x + y < 9 Step 1: Choose one variable to eliminate. Step 2: Choose a multiplier for one or both equations, so that the Step 3: Add the two equations. One variable will be eliminated. Step 4: Solve for the other variable. Step 5: Replace the solved variable with its value. Solve for the remaining variable. SOLUTION REGION equations have opposite terms. Multiply by the multipliers.

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