Find A Coefficient A A A For The Linear Equation A X − Y = 4 A X - Y = 4 A X − Y = 4 Such That The Graph Of The Equation Passes Through The Point M ( 3 , 5 M(3,5 M ( 3 , 5 ]. Graph This Equation.

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Introduction

In mathematics, a linear equation is a type of equation that can be written in the form $ax + by = c$, where $a$, $b$, and $c$ are constants, and $x$ and $y$ are variables. The graph of a linear equation is a straight line that passes through the point $(0, c/a)$ and has a slope of $a/b$. In this article, we will discuss how to find a coefficient $a$ for the linear equation $ax - y = 4$ such that the graph of the equation passes through the point $M(3,5)$.

Understanding the Problem

To find the coefficient $a$, we need to substitute the coordinates of the point $M(3,5)$ into the equation $ax - y = 4$ and solve for $a$. This will give us the value of $a$ that makes the graph of the equation pass through the point $M(3,5)$.

Substituting the Coordinates of the Point $M(3,5)$

We will substitute the coordinates of the point $M(3,5)$ into the equation $ax - y = 4$ as follows:

$$a(3) - 5 = 4$$

Solving for $a$

Now, we will solve for $a$ by isolating it on one side of the equation:

$$3a - 5 = 4$$

$$3a = 4 + 5$$

$$3a = 9$$

$$a = 9/3$$

$$a = 3$$

Conclusion

Therefore, the coefficient $a$ for the linear equation $ax - y = 4$ such that the graph of the equation passes through the point $M(3,5)$ is $a = 3$.

Graphing the Equation

To graph the equation $3x - y = 4$, we can use the slope-intercept form of a linear equation, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Converting the Equation to Slope-Intercept Form

We will convert the equation $3x - y = 4$ to slope-intercept form by isolating $y$ on one side of the equation:

$$-y = -3x + 4$$

$$y = 3x - 4$$

Finding the Slope and Y-Intercept

The slope of the equation is $3$, and the y-intercept is $-4$.

Graphing the Equation

To graph the equation, we can use the slope-intercept form of a linear equation, which is $y = mx + b$. We will plot the point $(0, -4)$, which is the y-intercept, and then use the slope to find another point on the line. We will draw a line through the two points to graph the equation.

Final Answer

The coefficient $a$ for the linear equation $ax - y = 4$ such that the graph of the equation passes through the point $M(3,5)$ is $a = 3$. The graph of the equation is a straight line with a slope of $3$ and a y-intercept of $-4$.

Example Use Case

The linear equation $3x - y = 4$ can be used to model a variety of real-world situations, such as the cost of producing a product. For example, if the cost of producing a product is $3x$ dollars, where $x$ is the number of units produced, and the profit is $4$ dollars, then the equation $3x - y = 4$ can be used to model the situation.

Step-by-Step Solution

To find the coefficient $a$ for the linear equation $ax - y = 4$ such that the graph of the equation passes through the point $M(3,5)$, follow these steps:

1. Substitute the coordinates of the point $M(3,5)$ into the equation $ax - y = 4$.
2. Solve for $a$ by isolating it on one side of the equation.
3. Convert the equation to slope-intercept form by isolating $y$ on one side of the equation.
4. Find the slope and y-intercept of the equation.
5. Graph the equation using the slope-intercept form of a linear equation.

Tips and Tricks

• When substituting the coordinates of a point into an equation, make sure to use the correct coordinates.
• When solving for a variable, make sure to isolate it on one side of the equation.
• When converting an equation to slope-intercept form, make sure to isolate $y$ on one side of the equation.
• When graphing an equation, make sure to use the slope-intercept form of a linear equation.

Common Mistakes

• Substituting the wrong coordinates into an equation.
• Not isolating a variable on one side of the equation.
• Not converting an equation to slope-intercept form correctly.
• Not graphing an equation correctly.

Conclusion

In conclusion, finding the coefficient $a$ for the linear equation $ax - y = 4$ such that the graph of the equation passes through the point $M(3,5)$ involves substituting the coordinates of the point into the equation, solving for $a$, converting the equation to slope-intercept form, finding the slope and y-intercept, and graphing the equation. By following these steps and avoiding common mistakes, you can find the coefficient $a$ and graph the equation correctly.

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Introduction

In our previous article, we discussed how to find the coefficient $a$ for the linear equation $ax - y = 4$ such that the graph of the equation passes through the point $M(3,5)$. In this article, we will answer some frequently asked questions about finding the coefficient $a$ and graphing the equation.

Q: What is the coefficient $a$ for the linear equation $ax - y = 4$?

A: The coefficient $a$ for the linear equation $ax - y = 4$ is $3$.

Q: How do I find the coefficient $a$ for the linear equation $ax - y = 4$?

A: To find the coefficient $a$, substitute the coordinates of the point $M(3,5)$ into the equation $ax - y = 4$ and solve for $a$.

Q: What is the slope of the linear equation $ax - y = 4$?

A: The slope of the linear equation $ax - y = 4$ is $3$.

Q: What is the y-intercept of the linear equation $ax - y = 4$?

A: The y-intercept of the linear equation $ax - y = 4$ is $-4$.

Q: How do I graph the linear equation $ax - y = 4$?

A: To graph the linear equation $ax - y = 4$, use the slope-intercept form of a linear equation, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Q: What is the equation of the line that passes through the point $M(3,5)$?

A: The equation of the line that passes through the point $M(3,5)$ is $3x - y = 4$.

Q: How do I find the equation of the line that passes through two points?

A: To find the equation of the line that passes through two points, use the slope-intercept form of a linear equation and substitute the coordinates of the two points into the equation.

Q: What is the difference between the slope and the y-intercept?

A: The slope is the ratio of the change in $y$ to the change in $x$, while the y-intercept is the value of $y$ when $x$ is equal to $0$.

Q: How do I use the slope and y-intercept to graph a line?

A: To graph a line using the slope and y-intercept, plot the y-intercept and then use the slope to find another point on the line.

Q: What are some real-world applications of linear equations?

A: Linear equations have many real-world applications, such as modeling the cost of producing a product, the profit of a business, and the distance traveled by an object.

Q: How do I use linear equations to model real-world situations?

A: To use linear equations to model real-world situations, identify the variables and constants in the situation and then write an equation that represents the situation.

Q: What are some common mistakes to avoid when working with linear equations?

A: Some common mistakes to avoid when working with linear equations include substituting the wrong coordinates into an equation, not isolating a variable on one side of the equation, and not converting an equation to slope-intercept form correctly.

Q: How do I check my work when solving a linear equation?

A: To check your work when solving a linear equation, substitute the solution back into the original equation and verify that it is true.

Q: What are some tips for graphing linear equations?

A: Some tips for graphing linear equations include using the slope-intercept form of a linear equation, plotting the y-intercept, and using the slope to find another point on the line.

Q: How do I use technology to graph linear equations?

A: To use technology to graph linear equations, use a graphing calculator or a computer program that can graph linear equations.

Q: What are some common applications of linear equations in science and engineering?

A: Linear equations have many applications in science and engineering, such as modeling the motion of objects, the flow of fluids, and the behavior of electrical circuits.

Q: How do I use linear equations to model the motion of an object?

A: To use linear equations to model the motion of an object, identify the variables and constants in the situation and then write an equation that represents the situation.

Q: What are some common mistakes to avoid when working with linear equations in science and engineering?

A: Some common mistakes to avoid when working with linear equations in science and engineering include substituting the wrong coordinates into an equation, not isolating a variable on one side of the equation, and not converting an equation to slope-intercept form correctly.

Conclusion

In conclusion, finding the coefficient $a$ for the linear equation $ax - y = 4$ and graphing the equation involves substituting the coordinates of the point $M(3,5)$ into the equation, solving for $a$, converting the equation to slope-intercept form, finding the slope and y-intercept, and graphing the equation. By following these steps and avoiding common mistakes, you can find the coefficient $a$ and graph the equation correctly.