CH 2 TESTName: ____________Solve The Following Equation:2) Y = − 1.5 X + 3.5 Y = -1.5x + 3.5 Y = − 1.5 X + 3.5

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a simple linear equation, $y = -1.5x + 3.5$. We will break down the solution into manageable steps, making it easy for readers to understand and apply the concept.

What is a Linear Equation?

A linear equation is an equation in which the highest power of the variable (in this case, $x$) is 1. It can be written in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The slope represents the rate of change of the variable, while the y-intercept represents the point where the line intersects the y-axis.

Understanding the Given Equation

The given equation is $y = -1.5x + 3.5$. Here, the slope ($m$) is $-1.5$, and the y-intercept ($b$) is $3.5$. To solve this equation, we need to find the value of $x$ that satisfies the equation.

Step 1: Isolate the Variable

To solve the equation, we need to isolate the variable $x$. We can do this by subtracting $3.5$ from both sides of the equation:

$$y - 3.5 = -1.5x$$

Step 2: Get the Variable on One Side

Next, we need to get the variable $x$ on one side of the equation. We can do this by dividing both sides of the equation by $-1.5$:

$$\frac{y - 3.5}{-1.5} = x$$

Step 3: Simplify the Equation

Now that we have isolated the variable $x$, we can simplify the equation by combining like terms:

$$x = \frac{y - 3.5}{-1.5}$$

Step 4: Substitute a Value for $y$

To find the value of $x$, we need to substitute a value for $y$. Let's say we want to find the value of $x$ when $y = 2$. We can substitute this value into the equation:

$$x = \frac{2 - 3.5}{-1.5}$$

Step 5: Simplify the Equation

Now that we have substituted a value for $y$, we can simplify the equation:

$$x = \frac{-1.5}{-1.5}$$

Step 6: Solve for $x$

Finally, we can solve for $x$ by simplifying the equation:

$$x = 1$$

Conclusion

In this article, we solved a simple linear equation, $y = -1.5x + 3.5$. We broke down the solution into manageable steps, making it easy for readers to understand and apply the concept. By following these steps, readers can solve linear equations with ease.

Tips and Tricks

• Always start by isolating the variable.
• Get the variable on one side of the equation by adding or subtracting the same value to both sides.
• Simplify the equation by combining like terms.
• Substitute a value for the variable to find the solution.
• Solve for the variable by simplifying the equation.

Common Mistakes to Avoid

• Not isolating the variable.
• Not getting the variable on one side of the equation.
• Not simplifying the equation.
• Not substituting a value for the variable.
• Not solving for the variable.

Real-World Applications

Linear equations have many real-world applications, including:

• Physics: Linear equations are used to describe the motion of objects.
• Engineering: Linear equations are used to design and optimize systems.
• Economics: Linear equations are used to model economic systems.
• Computer Science: Linear equations are used in algorithms and data structures.

Conclusion

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (in this case, $x$) is 1. It can be written in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Q: How do I solve a linear equation?

A: To solve a linear equation, follow these steps:

1. Isolate the variable by adding or subtracting the same value to both sides of the equation.
2. Get the variable on one side of the equation by adding or subtracting the same value to both sides.
3. Simplify the equation by combining like terms.
4. Substitute a value for the variable to find the solution.
5. Solve for the variable by simplifying the equation.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable (in this case, $x$) is 1. A quadratic equation is an equation in which the highest power of the variable (in this case, $x$) is 2. For example, $y = 2x + 3$ is a linear equation, while $y = 2x^2 + 3x + 4$ is a quadratic equation.

Q: How do I graph a linear equation?

A: To graph a linear equation, follow these steps:

1. Find the y-intercept by setting $x = 0$ and solving for $y$.
2. Find the slope by dividing the change in $y$ by the change in $x$.
3. Plot the y-intercept on the graph.
4. Plot a second point on the graph using the slope and the y-intercept.
5. Draw a line through the two points to graph the equation.

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

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

Q: How do I find the slope of a linear equation?

A: To find the slope of a linear equation, divide the change in $y$ by the change in $x$. For example, if the equation is $y = 2x + 3$, the slope is 2.

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

A: The y-intercept of a linear equation is the point where the line intersects the y-axis. To find the y-intercept, set $x = 0$ and solve for $y$.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, follow these steps:

1. Write the equations in the form $y = mx + b$.
2. Graph the equations on the same coordinate plane.
3. Find the point of intersection of the two lines.
4. Write the solution as an ordered pair (x, y).

Q: What is the difference between a linear equation and a nonlinear equation?

A: A linear equation is an equation in which the highest power of the variable (in this case, $x$) is 1. A nonlinear equation is an equation in which the highest power of the variable (in this case, $x$) is greater than 1. For example, $y = 2x + 3$ is a linear equation, while $y = 2x^2 + 3x + 4$ is a nonlinear equation.

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By following the steps outlined in this article, readers can solve linear equations with ease. Remember to always isolate the variable, get the variable on one side of the equation, simplify the equation, substitute a value for the variable, and solve for the variable. With practice and patience, readers can become proficient in solving linear equations and apply the concept to real-world problems.