Find an equation of the line that has slope 2 and passes through (1, -3).
Understand the Problem
The question is asking for the equation of a line that has a specific slope and passes through a given point. The approach requires applying the point-slope form of a linear equation.
Answer
The equation of the line is \( y = 2x - 5 \).
Answer for screen readers
The equation of the line is: $$ y = 2x - 5 $$
Steps to Solve
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Identify the point-slope form The point-slope form of a linear equation is given by the formula: $$ y - y_1 = m(x - x_1) $$ where ( m ) is the slope and ( (x_1, y_1) ) is the given point.
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Substitute the values Here, the slope ( m = 2 ) and the point ( (x_1, y_1) = (1, -3) ). Substitute these values into the point-slope formula: $$ y - (-3) = 2(x - 1) $$
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Simplify the equation Now simplify the equation. First, eliminate the parentheses: $$ y + 3 = 2(x - 1) $$ This simplifies to: $$ y + 3 = 2x - 2 $$
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Isolate ( y ) To rewrite the equation in slope-intercept form ( y = mx + b ), subtract 3 from both sides: $$ y = 2x - 2 - 3 $$ which simplifies to: $$ y = 2x - 5 $$
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Convert to standard form (optional) If desired, we can convert this to standard form ( Ax + By + C = 0 ) by rearranging: $$ 2x - y - 5 = 0 $$
The equation of the line is: $$ y = 2x - 5 $$
More Information
The equation represents a straight line with a slope of 2, indicating that for every unit increase in ( x ), ( y ) increases by 2 units. The line passes through the point ( (1, -3) ).
Tips
- Forgetting to switch signs when moving terms across the equation can lead to errors.
- Confusing the point-slope form with the slope-intercept form.
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