write the equation of the line represented by y = 2/3x – 5 in standard form.
The correct answer and explanation is:
Correct Answer:
The equation of the line y=23x−5y = \frac{2}{3}x – 5 in standard form is: 2x−3y=152x – 3y = 15
Explanation (300 words):
To convert a linear equation from slope-intercept form to standard form, we need to understand both formats:
- Slope-intercept form: y=mx+by = mx + b, where mm is the slope and bb is the y-intercept.
- Standard form: Ax+By=CAx + By = C, where AA, BB, and CC are integers, and typically AA is positive.
We start with the given equation: y=23x−5y = \frac{2}{3}x – 5
Step 1: Eliminate the fraction
Multiply every term by the denominator (3) to eliminate the fraction: 3y=2x−153y = 2x – 15
Step 2: Rearrange to standard form Ax+By=CAx + By = C
We want the xx and yy terms on the left side and the constant on the right: −2x+3y=−15-2x + 3y = -15
Step 3: Make AA positive
Standard form usually requires AA (the coefficient of xx) to be a positive integer. So, multiply the entire equation by -1: 2x−3y=152x – 3y = 15
Now we have the equation in standard form.
Why this matters:
Standard form is useful in algebra for solving systems of equations (e.g., using elimination or substitution), and in applications like graphing lines or finding intercepts quickly. For example:
- xx-intercept: Set y=0y = 0 → 2x=152x = 15 → x=7.5x = 7.5
- yy-intercept: Set x=0x = 0 → −3y=15-3y = 15 → y=−5y = -5
Thus, rewriting equations in standard form gives us flexibility in both algebraic operations and real-world applications.