10th Grade Linear Equations in Two Variables: Standard Form and Slope

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Questions and Answers

In which form is it easiest to identify the slope and y-intercept of a linear equation?

  • Point-slope form: $y - y_1 = m(x - x_1)$
  • Standard form: $ax + by = c$
  • General form: $ax + by + c = 0$
  • Slope-intercept form: $y = mx + c$ (correct)

What does a negative slope indicate in the context of a linear equation?

  • The line is increasing from left to right.
  • The line is decreasing from left to right. (correct)
  • The line is vertical.
  • The line is horizontal.

What is the standard form of a linear equation in two variables?

  • $y = ax^2 + bx + c$
  • $x + y = k$
  • $y = mx + c$
  • $ax + by = c$ (correct)

What is the general form of a quadratic equation?

<p>$ax^2 + bx + c = 0$ (C)</p>
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What does the discriminant of a quadratic equation determine?

<p>The type and number of solutions (A)</p>
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What is the quadratic formula used for?

<p>Finding the roots of a quadratic equation (D)</p>
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Study Notes

Linear Equations

  • The slope and y-intercept of a linear equation are easiest to identify in slope-intercept form (y = mx + b).
  • A negative slope indicates a downward slope, meaning as the input increases, the output decreases.

Standard Form of Linear Equations

  • The standard form of a linear equation in two variables is Ax + By = C, where A, B, and C are integers.

Quadratic Equations

  • The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants.
  • The discriminant (b^2 - 4ac) of a quadratic equation determines the number of solutions:
    • Positive discriminant indicates two distinct solutions.
    • Zero discriminant indicates one solution.
    • Negative discriminant indicates no real solutions.
  • The quadratic formula (x = (-b ± √(b^2 - 4ac)) / 2a) is used to find the solutions of a quadratic equation.

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