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Algebra I – Equations with Two Variables

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    Study Guide Algebra I – Equa Ɵ ons with Two Variables 1. Solving Systems of Equa Ɵ ons (Simultaneous Equa Ɵ ons) What Is a System of Equations? Sometimes, you are given two different equations with the same two unknowns (usually (x) and (y)). Your goal is to find values for both variables that make both equations true at the same time . This is called solving a system of equations or simultaneous equations . There are three common methods to solve them: 1. Addition/Subtraction (Elimination) Method 2. Substitution Method 3. Graphing Method In this chapter, we will focus on the Addition/Subtraction Method , also known as the Elimination Method . The Addition/Subtraction (Elimination) Method The elimination method works by removing one variable so you can solve for the other more easily. The idea is simple: • Make the coefficients (numbers in front of variables) match or be opposites. • Add or subtract the equations. • One variable disappears (is eliminated). • Solve what remains.

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    Study Guide Steps to Use the Elimination Method Follow these steps carefully: Step 1: Make the Coefficients Match Multiply one or both equations (if needed) so that: • The numbers in front of one variable are the same, or • They are exact opposites. This allows that variable to cancel out when adding or subtracting. Step 2: Add or Subtract the Equations Add (or subtract) the two equations. One variable will be eliminated. Step 3: Solve for the Remaining Variable Now you will have an equation with only one unknown. Solve it. Step 4: Substitute Back Take the value you just found and substitute it into one of the original equations. Then solve for the other variable. Example 1 Solve for (x) and (y):

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    Study Guide Step 1: Add the Equations Notice that: • One equation has +y • The other has – y That means if we add the equations, the (y) terms will cancel. Add them: Step 2: Solve for (x) Divide both sides by 2: Step 3: Substitute Back to Find (y) Now substitute (x = 5) into one of the original equations. Use:

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    Study Guide Substitute: Subtract 5 from both sides: Final Answer Always Check Your Answer To make sure your solution is correct, substitute (x = 5) and (y = 2) into both original equations. Check: 1. (5 + 2 = 7) 2. (5 - 2 = 3) Both equations are true. So the solution is correct. Answer: x = 5, y = 2 Important Note In this example, we found one unique solution . However, not all systems behave this way.

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    Study Guide Sometimes: • There is no solution . • There are infinitely many solutions . When using the elimination method, always check your final result carefully. Example 2 Solve for (x) and (y): Step 1: Make the Coefficients Match Look at the y - terms : • First equation has 3y • Second equation has y To eliminate (y), we want both equations to have the same coefficient in front of (y). So we multiply the second equation by 3 : This gives: Now the system looks like this:

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    Study Guide Now both equations have 3y . Perfect! Step 2: Subtract the Equations Since both equations have +3y , subtract one equation from the other to eliminate (y). Subtract the second equation from the first: The (y) - terms disappear! Step 3: Solve for (x) Divide both sides by – 3: Step 4: Substitute Back to Find (y) Now substitute (x = 5) into one of the original equations. Use:

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    Study Guide Substitute: Subtract 10 from both sides: Final Answer Check Your Solution Substitute (x = 5) and (y = 3) into both equations. First equation: Second equation: Both equations are true. So the solution is correct! Important Reminder If the coefficients are already the same (or opposites), you do not need to multiply anything first.

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    Study Guide You can simply add or subtract. Example 3 Solve for (a) and (b): Step 1: Look Carefully Notice something interesting. If we multiply the first equation by 2: We get: That is exactly the second equation ! What Does This Mean? This tells us something very important: The two equations are actually the same equation written in two different ways . If we subtract one from the other, we get:

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    Study Guide This statement is always true . What Type of Solution Is This? In this case: • The system does not have one unique solution. • It has infinitely many solutions . Why? Because any values of (a) and (b) that satisfy one equation will automatically satisfy the other. For example: If (a = - 6) and (b = 5): And: Both equations are true. Example 4 Solve for (x) and (y): Step 1: Try Eliminating a Variable Let’s try multiplying the first equation by 2:

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    Study Guide This gives: Now compare the two equations: They look almost identical — but the numbers on the right side are different! Step 2: Subtract the Equations If we subtract one equation from the other: This statement is never true . What Does This Mean? When solving systems, if you end up with something impossible like: it means the system has no solution . These two lines would never meet on a graph. They are parallel. Important Idea

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