Creating a two-variable linear equation involves identifying a linear relationship between two variables, usually referred to as \( x \) and \( y \). Here's how you can create one step-by-step:
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Understand the Format: A two-variable linear equation can be expressed in the standard form as: \[ Ax + By = C \] or in slope-intercept form: \[ y = mx + b \] where:
- \( A \), \( B \), and \( C \) are constants.
- \( m \) is the slope of the line.
- \( b \) is the y-intercept (the value of \( y \) when \( x = 0 \)).
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Define Your Variables: Decide what your two variables will represent. For example, let \( x \) represent the number of hours studied, and \( y \) represent the score on an exam.
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Determine a Relationship: Establish a linear relationship between the two variables. For instance, you might say that for every hour studied, the exam score increases by 10 points, starting from a base score of 50 when no hours are studied.
From this assumption, you can derive the equation.
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Write Your Equation:
- If the increase in score is 10 points per hour studied, you can express this relationship as: \[ y = 10x + 50 \]
- In standard form, this would be: \[ -10x + y = 50 \quad \text{or rearranging gives } 10x - y = -50 \]
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Verify the Equation: Ensure that the equation represents the intended relationship:
- If \( x = 0 \) (no hours studied), then \( y = 50 \) (base score).
- If \( x = 2 \) (two hours studied), then \( y = 10(2) + 50 = 70 \), which indicates a 20-point increase, consistent with our original relationship.
Now you have a two-variable linear equation. You can also create different equations by adjusting the slope and y-intercept according to different scenarios or relationships.