Find The Equation Of The Line With A Slope Of $1$ That Passes Through The Point $(1, -4$\]. Write Your Equation In The Form $y = Mx + B$.$y = \square$---Calculate The Equation Of The Line Going Through The Points

Understanding the Problem

To find the equation of a line with a slope of $1$ that passes through the point $(1, -4)$, we need to use the point-slope form of a linear equation, which is given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the given point and $m$ is the slope.

Recalling the Point-Slope Form

The point-slope form of a linear equation is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is the given point and $m$ is the slope.

Substituting the Given Values

We are given that the slope $m = 1$ and the point $(x_1, y_1) = (1, -4)$. Substituting these values into the point-slope form, we get:

$$y - (-4) = 1(x - 1)$$

Simplifying the Equation

Simplifying the equation, we get:

$$y + 4 = x - 1$$

Rearranging the Equation

Rearranging the equation to the form $y = mx + b$, we get:

$$y = x - 1 - 4$$

Simplifying the Equation

Simplifying the equation, we get:

$$y = x - 5$$

Conclusion

Therefore, the equation of the line with a slope of $1$ that passes through the point $(1, -4)$ is $y = x - 5$.

Interpreting the Equation

The equation $y = x - 5$ represents a line with a slope of $1$ that passes through the point $(1, -4)$. This means that for every unit increase in the x-coordinate, the y-coordinate increases by one unit. The y-intercept of the line is $-5$, which means that the line intersects the y-axis at the point $(0, -5)$.

Graphing the Line

To graph the line, we can plot the point $(1, -4)$ and then use the slope to find other points on the line. Since the slope is $1$, we can move one unit to the right and one unit up from the point $(1, -4)$ to find the point $(2, -3)$. We can continue this process to find other points on the line.

Using the Equation to Find Other Points

We can use the equation $y = x - 5$ to find other points on the line. For example, if we want to find the point on the line that has an x-coordinate of $3$, we can substitute $x = 3$ into the equation:

$$y = 3 - 5$$

$$y = -2$$

Therefore, the point on the line with an x-coordinate of $3$ is $(3, -2)$.

Using the Equation to Find the Y-Intercept

We can use the equation $y = x - 5$ to find the y-intercept of the line. The y-intercept is the point where the line intersects the y-axis, which means that the x-coordinate is $0$. We can substitute $x = 0$ into the equation:

$$y = 0 - 5$$

$$y = -5$$

Therefore, the y-intercept of the line is $-5$.

Using the Equation to Find the X-Intercept

We can use the equation $y = x - 5$ to find the x-intercept of the line. The x-intercept is the point where the line intersects the x-axis, which means that the y-coordinate is $0$. We can substitute $y = 0$ into the equation:

$$0 = x - 5$$

$$x = 5$$

Therefore, the x-intercept of the line is $5$.

Conclusion

In conclusion, the equation of the line with a slope of $1$ that passes through the point $(1, -4)$ is $y = x - 5$. We can use this equation to find other points on the line, the y-intercept, and the x-intercept.

Q: What is the point-slope form of a linear equation?

A: The point-slope form of a linear equation is given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the given point and $m$ is the slope.

Q: How do I find the equation of a line with a given slope and point?

A: To find the equation of a line with a given slope and point, you can use the point-slope form of a linear equation. Simply substitute the given values into the equation and simplify.

Q: What is the slope-intercept form of a linear equation?

A: The slope-intercept form of a linear equation is given by $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Q: How do I convert the point-slope form to the slope-intercept form?

A: To convert the point-slope form to the slope-intercept form, you can simplify the equation by combining like terms and isolating the y-variable.

Q: What is the y-intercept of a line?

A: The y-intercept of a line is the point where the line intersects the y-axis, which means that the x-coordinate is $0$.

Q: How do I find the y-intercept of a line?

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

Q: What is the x-intercept of a line?

A: The x-intercept of a line is the point where the line intersects the x-axis, which means that the y-coordinate is $0$.

Q: How do I find the x-intercept of a line?

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

Q: Can I use the equation of a line to find other points on the line?

A: Yes, you can use the equation of a line to find other points on the line. Simply substitute different values of $x$ into the equation and solve for $y$.

Q: Can I use the equation of a line to find the slope of the line?

A: No, you cannot use the equation of a line to find the slope of the line. The slope is a given value that is used to find the equation of the line.

Q: Can I use the equation of a line to find the y-intercept of the line?

A: Yes, you can use the equation of a line to find the y-intercept of the line. Simply substitute $x = 0$ into the equation and solve for $y$.

Q: Can I use the equation of a line to find the x-intercept of the line?

A: Yes, you can use the equation of a line to find the x-intercept of the line. Simply substitute $y = 0$ into the equation and solve for $x$.

Q: What is the difference between the point-slope form and the slope-intercept form of a linear equation?

A: The point-slope form of a linear equation is given by $y - y_1 = m(x - x_1)$, while the slope-intercept form is given by $y = mx + b$. The point-slope form is used to find the equation of a line with a given point and slope, while the slope-intercept form is used to find the equation of a line with a given slope and y-intercept.

Q: Can I use the equation of a line to find the equation of a line with a different slope?

A: No, you cannot use the equation of a line to find the equation of a line with a different slope. The equation of a line is specific to the given slope and point, and cannot be used to find the equation of a line with a different slope.

Q: Can I use the equation of a line to find the equation of a line with a different point?

A: No, you cannot use the equation of a line to find the equation of a line with a different point. The equation of a line is specific to the given point and slope, and cannot be used to find the equation of a line with a different point.

Q: What is the relationship between the slope and the y-intercept of a line?

A: The slope and the y-intercept of a line are related by the equation $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The slope determines the steepness of the line, while the y-intercept determines the position of the line on the y-axis.

Q: Can I use the equation of a line to find the equation of a line with a different y-intercept?

A: Yes, you can use the equation of a line to find the equation of a line with a different y-intercept. Simply substitute a different value for $b$ into the equation $y = mx + b$.

Q: Can I use the equation of a line to find the equation of a line with a different slope?

A: Yes, you can use the equation of a line to find the equation of a line with a different slope. Simply substitute a different value for $m$ into the equation $y = mx + b$.

Q: What is the relationship between the x-intercept and the y-intercept of a line?

A: The x-intercept and the y-intercept of a line are related by the equation $x = -\frac{b}{m}$, where $b$ is the y-intercept and $m$ is the slope.

Q: Can I use the equation of a line to find the equation of a line with a different x-intercept?

A: Yes, you can use the equation of a line to find the equation of a line with a different x-intercept. Simply substitute a different value for $x$ into the equation $x = -\frac{b}{m}$.

Q: Can I use the equation of a line to find the equation of a line with a different y-intercept?

A: Yes, you can use the equation of a line to find the equation of a line with a different y-intercept. Simply substitute a different value for $b$ into the equation $x = -\frac{b}{m}$.

Q: What is the relationship between the slope and the x-intercept of a line?

A: The slope and the x-intercept of a line are related by the equation $x = -\frac{b}{m}$, where $b$ is the y-intercept and $m$ is the slope.

Q: Can I use the equation of a line to find the equation of a line with a different slope and x-intercept?

A: Yes, you can use the equation of a line to find the equation of a line with a different slope and x-intercept. Simply substitute different values for $m$ and $x$ into the equation $x = -\frac{b}{m}$.

Q: Can I use the equation of a line to find the equation of a line with a different slope and y-intercept?

A: Yes, you can use the equation of a line to find the equation of a line with a different slope and y-intercept. Simply substitute different values for $m$ and $b$ into the equation $y = mx + b$.

Q: What is the relationship between the x-intercept and the y-intercept of a line?

A: The x-intercept and the y-intercept of a line are related by the equation $x = -\frac{b}{m}$, where $b$ is the y-intercept and $m$ is the slope.

Q: Can I use the equation of a line to find the equation of a line with a different x-intercept and y-intercept?

A: Yes, you can use the equation of a line to find the equation of a line with a different x-intercept and y-intercept. Simply substitute different values for $x$ and $b$ into the equation $x = -\frac{b}{m}$.

Q: What is the relationship between the slope and the x-intercept of a line?

A: The slope and the x-intercept of a line are related by the equation $x = -\frac{b}{m}$, where $b$ is the y-intercept and $m$ is the slope.

Q: Can I use the equation of a line to find the equation of a line with a different slope and x-intercept?

A: Yes, you can use the equation of a line to find the equation of a line with a different slope and x-intercept. Simply substitute different values for $m$ and $x$ into the equation $x = -\frac{b}{m}$.

Q: Can I use the equation of a line to find the equation of a line with a different slope and y-intercept?

A: Yes, you can use the equation of a line to find the equation of a line with a different slope and y-intercept. Simply substitute different values for $m$ and $b$ into the equation $y = mx + b$.

**Q: What is the relationship between the x-intercept and the y