Determine The Intercepts Of The Line. Do Not Round Your Answers.Equation: $y + 1 = 3(x - 4$\]- $x$-intercept: $(\square, \square$\]- $y$-intercept: $(\square, \square$\]

Understanding the Equation of a Line

The equation of a line can be written in various forms, including the slope-intercept form, point-slope form, and standard form. In this article, we will focus on determining the intercepts of a line given in the standard form. The standard form of a line is represented as $Ax + By = C$, where $A$, $B$, and $C$ are constants.

The Given Equation

The given equation of the line is $y + 1 = 3(x - 4)$. To determine the intercepts, we need to rewrite the equation in the standard form.

Rewriting the Equation in Standard Form

To rewrite the equation in standard form, we need to expand the right-hand side of the equation.

y + 1 = 3x - 12

Next, we need to isolate the terms involving $y$ on one side of the equation.

y = -3x + 13

Now, we can rewrite the equation in standard form by multiplying both sides of the equation by $-1$.

3x - y = 13

Determining the $x$-Intercept

The $x$-intercept is the point on the line where $y = 0$. To determine the $x$-intercept, we need to substitute $y = 0$ into the equation and solve for $x$.

3x - 0 = 13

Simplifying the equation, we get:

3x = 13

Dividing both sides of the equation by $3$, we get:

x = \frac{13}{3}

Therefore, the $x$-intercept is $\left(\frac{13}{3}, 0\right)$.

Determining the $y$-Intercept

The $y$-intercept is the point on the line where $x = 0$. To determine the $y$-intercept, we need to substitute $x = 0$ into the equation and solve for $y$.

3(0) - y = 13

Simplifying the equation, we get:

-y = 13

Multiplying both sides of the equation by $-1$, we get:

y = -13

Therefore, the $y$-intercept is $\left(0, -13\right)$.

Conclusion

In this article, we determined the intercepts of the line given in the standard form. We first rewrote the equation in standard form, then determined the $x$-intercept by substituting $y = 0$ into the equation and solving for $x$. Finally, we determined the $y$-intercept by substituting $x = 0$ into the equation and solving for $y$. The $x$-intercept is $\left(\frac{13}{3}, 0\right)$, and the $y$-intercept is $\left(0, -13\right)$.

Key Takeaways

• The standard form of a line is represented as $Ax + By = C$.
• To determine the intercepts of a line, we need to rewrite the equation in standard form.
• The $x$-intercept is the point on the line where $y = 0$.
• The $y$-intercept is the point on the line where $x = 0$.

Practice Problems

1. Determine the intercepts of the line given by the equation $2x + 3y = 6$.
2. Determine the intercepts of the line given by the equation $x - 2y = 3$.
3. Determine the intercepts of the line given by the equation $y = 2x - 1$.

Solutions

1. The equation can be rewritten as $y = -\frac{2}{3}x + 2$. The $x$-intercept is $\left(\frac{3}{2}, 0\right)$, and the $y$-intercept is $\left(0, 2\right)$.
2. The equation can be rewritten as $y = \frac{1}{2}x - \frac{3}{2}$. The $x$-intercept is $\left(6, 0\right)$, and the $y$-intercept is $\left(0, -\frac{3}{2}\right)$.
3. The equation is already in slope-intercept form. The $x$-intercept is $\left(1, 0\right)$, and the $y$-intercept is $\left(0, -1\right)$.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart
• [3] "Precalculus" by Michael Sullivan
  Frequently Asked Questions (FAQs) =====================================

Q: What is the standard form of a line?

A: The standard form of a line is represented as $Ax + By = C$, where $A$, $B$, and $C$ are constants.

Q: How do I determine the intercepts of a line?

A: To determine the intercepts of a line, you need to rewrite the equation in standard form. Then, substitute $y = 0$ to find the $x$-intercept and substitute $x = 0$ to find the $y$-intercept.

Q: What is the $x$-intercept?

A: The $x$-intercept is the point on the line where $y = 0$. It is the value of $x$ when the line crosses the $x$-axis.

Q: What is the $y$-intercept?

A: The $y$-intercept is the point on the line where $x = 0$. It is the value of $y$ when the line crosses the $y$-axis.

Q: How do I find the $x$-intercept?

A: To find the $x$-intercept, substitute $y = 0$ into the equation and solve for $x$.

Q: How do I find the $y$-intercept?

A: To find the $y$-intercept, substitute $x = 0$ into the equation and solve for $y$.

Q: What if the equation is not in standard form?

A: If the equation is not in standard form, you need to rewrite it in standard form before determining the intercepts.

Q: Can I use a graphing calculator to find the intercepts?

A: Yes, you can use a graphing calculator to find the intercepts. However, it is also important to understand the mathematical concepts behind finding the intercepts.

Q: What are some common mistakes to avoid when finding intercepts?

A: Some common mistakes to avoid when finding intercepts include:

• Not rewriting the equation in standard form
• Not substituting the correct value for $y$ or $x$
• Not solving for the correct variable
• Not checking the units of the variables

Q: How do I check my work when finding intercepts?

A: To check your work when finding intercepts, make sure to:

• Rewrite the equation in standard form
• Substitute the correct value for $y$ or $x$
• Solve for the correct variable
• Check the units of the variables
• Graph the equation to verify the intercepts

Conclusion

In this article, we answered some frequently asked questions about finding the intercepts of a line. We covered topics such as the standard form of a line, determining the intercepts, and common mistakes to avoid. We also provided some tips for checking your work when finding intercepts. By following these tips and understanding the mathematical concepts behind finding intercepts, you can become more confident and proficient in solving problems involving linear equations.

Practice Problems

1. Determine the intercepts of the line given by the equation $2x + 3y = 6$.
2. Determine the intercepts of the line given by the equation $x - 2y = 3$.
3. Determine the intercepts of the line given by the equation $y = 2x - 1$.

Solutions

1. The equation can be rewritten as $y = -\frac{2}{3}x + 2$. The $x$-intercept is $\left(\frac{3}{2}, 0\right)$, and the $y$-intercept is $\left(0, 2\right)$.
2. The equation can be rewritten as $y = \frac{1}{2}x - \frac{3}{2}$. The $x$-intercept is $\left(6, 0\right)$, and the $y$-intercept is $\left(0, -\frac{3}{2}\right)$.
3. The equation is already in slope-intercept form. The $x$-intercept is $\left(1, 0\right)$, and the $y$-intercept is $\left(0, -1\right)$.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart
• [3] "Precalculus" by Michael Sullivan