Solving 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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How do you determine the number of solutions in a system of equations?

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The number of solutions can be determined by analyzing the slopes of the lines represented by the equations: 1) One solution (intersecting lines), 2) No solution (parallel lines), 3) Infinitely many solutions (same line).

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What is the slope-intercept form of a linear equation?

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The slope-intercept form is given by the equation y = mx + b, where m is the slope and b is the y-intercept.

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How do you convert standard form to slope-intercept form?

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To convert from standard form (Ax + By = C) to slope-intercept form (y = mx + b), solve for y in terms of x.

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What does it mean for two lines to be parallel?

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Two lines are parallel if they have the same slope but different y-intercepts, meaning they will never intersect.

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What is the significance of the point of intersection in a system of equations?

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The point of intersection represents the solution to the system, indicating the values of the variables that satisfy all equations in the system.

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How can you solve a system of equations graphically?

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To solve graphically, plot each equation on the same coordinate plane and identify the point(s) where the lines intersect.

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What is the elimination method for solving systems of equations?

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

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What is the substitution method for solving systems of equations?

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

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How do you interpret the solution of a system of equations in a real-world context?

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The solution can represent quantities or values that satisfy the conditions of the problem, such as the number of items sold or the amount of resources used.