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Algebra II - Linear Sentences in Two Variables

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    Study Guide Algebra II – Linear Sentences in Two Variables 1. Linear Equa Ɵ ons: Solu Ɵ ons Using Graphing with Two Variables Sometimes you are given two linear equations and asked to find a solution that satisfies both equations at the same time . One way to do this is graphing . When you graph both equations on the same coordinate plane , the point where the two lines intersect (cross) represents the solution to the system . That intersection point gives the ordered pair (x, y) that works for both equations. Example 1: Solve Figure 1 Solve the system by graphing:

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    Study Guide Step 1: Graph Both Equations Plot each equation as a line on the same coordinate plane . • The first equation forms one line. • The second equation forms another line. Step 2: Find the Intersection Look for the point where the two lines cross . From the graph, the lines intersect at: So the solution to the system is : Step 3: Check the Solution To make sure the solution is correct, substitute (x = 3) and (y = - 2) into both equations. Check the First Equation Substitute the values: The equation is true ✔

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    Study Guide Check the Second Equation Substitute the values: This equation is also true ✔ Since the ordered pair works for both equations , it is the correct solution . Important Note About Graphing Graphing is helpful for understanding systems visually, but it has some limitations. • The intersection point must be easy to see on the graph . • It usually works best when the solution is close to the origin and has integer values . • Sometimes the exact intersection is hard to estimate just by looking at the graph. Because of this, graphing is often used less frequently than other methods (like substitution or elimination) when solving systems. Special Types of Systems When graphing systems of equations, two special situations can occur. 1. Dependent System A dependent system happens when the two equations represent the same line . This means the graphs completely overlap (coincide) . Since every point on the line satisfies both equations, the system has infinitely many solutions . In other words, the solution set can be written as all points on the line .

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    Study Guide 2. Inconsistent System An inconsistent system happens when the two lines are parallel . Parallel lines never intersect , so there is no solution . The solution set is written as: This symbol means the empty set , which means no solutions exist . Summary • A system of equations contains two or more equations with the same variables. • Graphing both equations on the same coordinate plane helps find the solution. • The intersection point of the two lines is the solution. • Always check the solution by substituting the values into both equations. Special Cases • Dependent system: Same line → infinitely many solutions • Inconsistent system: Parallel lines → no solution Example solution from the graph: This point satisfies both equations. 2. Quiz: Linear Equa Ɵ ons: Solu Ɵ ons Using Graphing with Two Variables 1. Question Solve the system of equations:

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    Study Guide Answer Choices • (x = - 3; y = 5) • (x = 3 ; y = - 5) • (x = 5 ; y = - 3) Correct Answer (x = 3 ; y = - 5) Why This Is Correct We solve the system using the elimination method . 1. Multiply the first equation by 2 so the (x) coefficients match: 2. Subtract the second equation: 3. Solve for (y): 4. Substitute (y = - 5) into (6x + 3y = 3): Therefore, the solution to the system is:

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    Study Guide 2. Question Solve the system of equations: Answer Choices • (x = 2 ; y = 7) • (x = - 2 ; y = 7) • (x = 3 ; y = 6) Correct Answer (x = 2 ; y = 7) Why This Is Correct First solve the second equation for (y): Next substitute this expression for (y) into the first equation:

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    Study Guide Now substitute (x = 2) back into (y = 17 – 5x): Therefore, the solution to the system is: 3. Question Solve the system of equations: Answer Choices  (x = 2; y = - 3)  (x = - 2; y = 3)  (x = - 2; y = - 3) Correct Answer (x = 2; y = - 3) Why This Is Correct Use the elimination method . 1. Multiply the first equation by 7 and the second equation by 2 to align the (x) terms:

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    Study Guide 2. Subtract the second equation from the first: 3. Solve for (y): 4. Substitute (y = - 3) into the first equation: But check with the second equation: This satisfies the equation, so the correct solution is: Therefore, the correct answer is: (x = - 2; y = - 3) 4. Question Solve the system of equations:

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    Study Guide Answer Choices • (x = - 3; y = 7) • (x = 3; y = - 7) • (x = 7; y = - 3) Correct Answer (x = - 3 ; y = 7) Why This Is Correct First rewrite the second equation in standard form: Now eliminate (x). Multiply the second equation by 5 and the first equation by 2 : Add the equations: Substitute (y = 7) into the first equation: Therefore, the solution to the system is:

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    Study Guide 5. Question Solve the system of equations: Answer Choices • (x = 3; y = - 5) • (x = 4; y = - 4) • (x = - 3; y = 5) Correct Answer (x = 4 ; y = - 4) Why This Is Correct First rewrite the second equation in standard form. Add (2y) to both sides: Now we have the system: Add the equations to eliminate (y):

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