What is the equation of a line, in slope-intercept form, that passes through the point (1, -2) and has a slope of 3?
Understand the Problem
The question is asking for the equation of a line in slope-intercept form (y = mx + b) that goes through a specific point (1, -2) with a given slope of 3. To find the correct equation, we will plug in the slope and the point into the slope-intercept form and solve for the y-intercept (b).
Answer
The equation of the line is \( y = 3x - 5 \).
Answer for screen readers
The equation of the line is ( y = 3x - 5 ).
Steps to Solve
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Identify the formula We are using the slope-intercept form of a line, which is given by the equation: $$ y = mx + b $$ Where ( m ) represents the slope and ( b ) is the y-intercept.
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Substitute known values Now, we substitute the known values into the equation. We have the slope ( m = 3 ) and a specific point ( (1, -2) ), meaning ( x = 1 ) and ( y = -2 ): $$ -2 = 3(1) + b $$
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Solve for b Next, we simplify the equation to isolate ( b ): $$ -2 = 3 + b $$ To find ( b ), we subtract 3 from both sides: $$ b = -2 - 3 $$ This simplifies to: $$ b = -5 $$
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Write the final equation Now that we have our slope ( m ) and the y-intercept ( b ), we can write the final equation of the line: $$ y = 3x - 5 $$
The equation of the line is ( y = 3x - 5 ).
More Information
In this problem, we used the slope-intercept form of a linear equation, which is a fundamental concept in algebra. This form is useful because it clearly shows the slope and the y-intercept, making it easy to graph the line.
Tips
- Forgetting to substitute the coordinates correctly into the equation. Always check that you're using ( x ) and ( y ) values properly based on the point given.
- Miscalculating the value of ( b ). It's crucial to carefully perform arithmetic operations when isolating ( b ).
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