Number of Solutions

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What is the definition of a linear equation?

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A linear equation is an equation of the first degree, meaning it has no exponents greater than one, and it can be written in the form ax + b = c, where a, b, and c are constants.

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How can you determine the number of solutions for a linear equation?

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The number of solutions for a linear equation can be determined by analyzing the relationship between the coefficients of the variables. If the equation simplifies to a true statement (e.g., 0 = 0), it has infinite solutions. If it simplifies to a false statement (e.g., 0 = 5), it has no solution. If it results in a unique value for the variable, it has one solution.

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What does it mean if an equation has one solution?

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An equation has one solution if there is exactly one value of the variable that satisfies the equation.

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What does it mean if an equation has no solution?

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An equation has no solution if there is no value of the variable that can satisfy the equation, often resulting in a contradiction.

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What does it mean if an equation has infinite solutions?

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An equation has infinite solutions if any value of the variable satisfies the equation, indicating that the two sides of the equation are equivalent for all values of the variable.

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What is the significance of the coefficients in a linear equation?

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The coefficients in a linear equation determine the slope and position of the line represented by the equation on a graph.

Front

How do you solve the equation 3x + 1 = 8 + 2x?

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To solve 3x + 1 = 8 + 2x, subtract 2x from both sides to get x + 1 = 8. Then, subtract 1 from both sides to find x = 7. This equation has one solution.

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How do you solve the equation 5x + 5 = 8 + 5x?

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To solve 5x + 5 = 8 + 5x, subtract 5x from both sides to get 5 = 8, which is a contradiction. Therefore, this equation has no solution.

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How do you solve the equation 2x + 4 = 4 + 2x?

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To solve 2x + 4 = 4 + 2x, subtract 2x from both sides to get 4 = 4, which is always true. Therefore, this equation has infinite solutions.

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How do you solve the equation 7x + 3 = 3x + 3?

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To solve 7x + 3 = 3x + 3, subtract 3x from both sides to get 4x + 3 = 3. Then, subtract 3 from both sides to find 4x = 0, leading to x = 0. This equation has one solution.