Systems of Equations

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What is a system of equations?

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A system of equations is a set of two or more equations with the same variables. The solution is the point(s) where the equations intersect.

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What does it mean for a point to be a solution to a system of equations?

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A point is a solution to a system of equations if it satisfies all equations in the system when substituted for the variables.

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How can you determine if (4, 1) is a solution for the system: y = -x + 5 and y = 2x - 7?

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Substitute x=4 and y=1 into both equations. For y = -x + 5: 1 = -4 + 5 (true). For y = 2x - 7: 1 = 2(4) - 7 (true). Thus, (4, 1) is a solution.

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What is the method of substitution in solving systems of equations?

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The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation.

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What is the method of elimination in solving systems of equations?

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The method of elimination involves adding or subtracting equations to eliminate one variable, making it easier to solve for the other variable.

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Solve the system: x - 4y = 16 and 5x + 4y = 8. What is the solution?

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The solution is (4, -3).

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Solve the system: x - 2y = -2 and 3x - 2y = 6. What is the solution?

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The solution is (4, 3).

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What does it mean if a system of equations has no solution?

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A system has no solution if the equations represent parallel lines that never intersect.

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What does it mean if a system of equations has infinitely many solutions?

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A system has infinitely many solutions if the equations represent the same line, meaning every point on the line is a solution.

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The solution is infinitely many solutions.