← Grade 8: Simultaneous Linear Equations
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Grade 8: Simultaneous Linear Equations
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Grade 8: Simultaneous Linear Equations — Flashcards
| Front | Back |
|---|---|
| What are simultaneous linear equations? | A pair of equations of the form Ax + By = C (A, B, C are specific numbers) where we seek values of x and y that make both equations true at the same time. |
| What is a "solution" to simultaneous linear equations? | The pair of values (x, y) that make both equations true at the same time. |
| Standard 8.EE.8a | Solve pairs of simultaneous linear equations; solutions correspond to points of intersection of their graphs, since intersection points satisfy both equations. |
| Tool: Add equals to equals | You can add the same quantity (or equal equations) to both sides of an equation without changing the solution (e.g., x=21, y=17 becomes x+y=38). |
| Tool: Subtract equals from equals | You can subtract the same quantity (or equal equations) from both sides without changing the solution (e.g., x=21, y=17 becomes x−y=4). |
| Tool: Multiply equals by a nonzero number | You can multiply both sides of an equation by the same nonzero number without changing the solution. |
| Tool: Substitution | Replace an expression in one equation with an equal expression obtained from the other equation. |
| Elimination by adding equations | Adding two simultaneous equations together can eliminate one variable, allowing you to solve for the other (e.g., adding x+y=38 and x−y=4 gives 2x=42). |
| Elimination by subtracting equations | Subtracting one simultaneous equation from another can eliminate one variable (e.g., subtracting x−y=4 from x+y=38 gives 2y=34). |
| How does substitution solve a word problem (Example 3)? | Write two equations from the given facts; use one equation to replace a variable in the other equation with an equal expression, reducing it to one equation in one variable. |
| Graphical meaning of a solution | The solution to a pair of simultaneous linear equations is the point (x, y) where the graphs of both lines intersect, since that point satisfies both equations. |
| Case 1: Different slopes | If two lines have different slopes (different rates of change), they are not parallel and intersect at exactly one point — a unique solution. |
| Case 2: Same slope, different intercepts | If two lines have the same slope but different y-intercepts, they are parallel and distinct — there is no solution. |
| Case 3: Same slope, same intercept | If two equations describe the same line (same slope and intercept), there are infinitely many solutions — every point on the line works. |
| How can you tell if two equations describe the same line? | One equation is a constant multiple of the other (equivalent expressions), producing identical slope and intercept. |
| Why must you always have two equations for two unknowns? | With only one equation (e.g., just knowing the sum of two numbers), there are infinitely many possible pairs; a second independent equation is needed to pin down a unique solution. |
| Example 4 setup | For 3x + y = 7 and x + 3y = 5, the slopes (−3 and −1/3) differ, so the lines intersect at one point: (2, 1), which satisfies both equations. |
| Example 6 setup | For 2x + 5y = 10 and 4x + 10y = 40, the lines have the same slope (−2/5) but different y-intercepts, so they are parallel with no solution. |
| Example 7 setup | For 2x + 5y = 20 and 4x + 10y = 40, both simplify to y = −(2/5)x + 4 — the same line — giving infinitely many solutions. |
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