Solve The Equation: Y − 2 = 2 ( X + 7 Y - 2 = 2(x + 7 Y − 2 = 2 ( X + 7 ]

Mar 13, 2025 by ADMIN 74 views

**Solving Linear Equations: A Step-by-Step Guide**

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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 specific type of linear equation, which is in the form of $y - 2 = 2(x + 7)$ . This equation involves a variable $y$ and a constant $2$ , and we need to isolate the variable $y$ to find its value.

Understanding the Equation

The given equation is $y - 2 = 2(x + 7)$ . To solve this equation, we need to isolate the variable $y$ on one side of the equation. The equation involves a constant $2$ on the right-hand side, which we need to eliminate to find the value of $y$ .

Step 1: Distribute the Constant

The first step in solving the equation is to distribute the constant $2$ to the terms inside the parentheses on the right-hand side. This will give us:

y - 2 = 2x + 14

Step 2: Add 2 to Both Sides

The next step is to add $2$ to both sides of the equation to eliminate the constant $-2$ on the left-hand side. This will give us:

y = 2x + 16

Step 3: Simplify the Equation

The equation is now in its simplest form, and we can see that the variable $y$ is isolated on the left-hand side. The equation is now in the form of $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

Interpretation of the Solution

The solution to the equation $y - 2 = 2(x + 7)$ is $y = 2x + 16$ . This means that for every value of $x$ , the corresponding value of $y$ is given by the equation $y = 2x + 16$ . This equation represents a linear relationship between the variables $x$ and $y$ .

Real-World Applications

Linear equations have numerous real-world applications in fields such as physics, engineering, economics, and computer science. For example, the equation $y = 2x + 16$ can be used to model the relationship between the distance traveled and the time taken in a linear motion.

Conclusion

Solving linear equations is an essential skill for students to master, and it has numerous real-world applications. In this article, we solved the equation $y - 2 = 2(x + 7)$ using a step-by-step approach. We distributed the constant, added $2$ to both sides, and simplified the equation to isolate the variable $y$ . The solution to the equation is $y = 2x + 16$ , which represents a linear relationship between the variables $x$ and $y$ .

Tips and Tricks

When solving linear equations, always start by distributing the constant to the terms inside the parentheses.
Add or subtract the same value to both sides of the equation to eliminate the constant.
Simplify the equation to isolate the variable on one side.
Use the equation to model real-world relationships between variables.

Common Mistakes

Failing to distribute the constant to the terms inside the parentheses.
Not adding or subtracting the same value to both sides of the equation.
Not simplifying the equation to isolate the variable.
Not using the equation to model real-world relationships between variables.

Practice Problems

Solve the equation $y + 3 = 2(x - 2)$ .
Solve the equation $y - 5 = 3(x + 1)$ .
Solve the equation $y + 2 = 4(x - 3)$ .

Solutions

$y = 2x - 1$
$y = 3x + 8$
$y = 4x - 10$

Conclusion

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Introduction

In our previous article, we discussed how to solve linear equations using a step-by-step approach. In this article, we will provide a Q&A guide to help students understand the concept of solving linear equations better.

Q1: What is a linear equation?

A linear equation is an equation in which the highest power of the variable(s) is 1. It can be written in the form of $ax + b = c$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable.

A1: How do I solve a linear equation?

To solve a linear equation, you need to isolate the variable on one side of the equation. You can do this by adding or subtracting the same value to both sides of the equation, or by multiplying or dividing both sides of the equation by the same non-zero value.

Q2: What is the difference between a linear equation and a quadratic equation?

A linear equation is an equation in which the highest power of the variable(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2. For example, $x + 2 = 3$ is a linear equation, while $x^2 + 4x + 4 = 0$ is a quadratic equation.

A2: How do I know if an equation is linear or quadratic?

To determine if an equation is linear or quadratic, you need to look at the highest power of the variable(s) in the equation. If the highest power is 1, the equation is linear. If the highest power is 2, the equation is quadratic.

Q3: What is the slope-intercept form of a linear equation?

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

A3: How do I convert a linear equation to slope-intercept form?

To convert a linear equation to slope-intercept form, you need to isolate the variable $y$ on one side of the equation. You can do this by adding or subtracting the same value to both sides of the equation, or by multiplying or dividing both sides of the equation by the same non-zero value.

Q4: What is the significance of the y-intercept in a linear equation?

The y-intercept in a linear equation represents the point at which the line intersects the y-axis. It is the value of $y$ when $x$ is equal to 0.

A4: How do I find the y-intercept of a linear equation?

To find the y-intercept of a linear equation, you need to set $x$ equal to 0 and solve for $y$ . This will give you the value of the y-intercept.

Q5: Can I solve a linear equation using a graph?

Yes, you can solve a linear equation using a graph. By plotting the equation on a coordinate plane, you can find the point at which the line intersects the x-axis or the y-axis.

A5: How do I graph a linear equation?

To graph a linear equation, you need to plot two points on the coordinate plane that satisfy the equation. You can then draw a line through the two points to represent the equation.

Conclusion

Solving linear equations is an essential skill for students to master, and it has numerous real-world applications. In this article, we provided a Q&A guide to help students understand the concept of solving linear equations better. We discussed the difference between linear and quadratic equations, the slope-intercept form of a linear equation, and how to find the y-intercept of a linear equation.

Tips and Tricks

Always start by distributing the constant to the terms inside the parentheses.
Add or subtract the same value to both sides of the equation to eliminate the constant.
Simplify the equation to isolate the variable on one side.
Use the equation to model real-world relationships between variables.

Common Mistakes

Failing to distribute the constant to the terms inside the parentheses.
Not adding or subtracting the same value to both sides of the equation.
Not simplifying the equation to isolate the variable.
Not using the equation to model real-world relationships between variables.

Practice Problems

Solve the equation $y + 3 = 2(x - 2)$ .
Solve the equation $y - 5 = 3(x + 1)$ .
Solve the equation $y + 2 = 4(x - 3)$ .

Solutions

$y = 2x - 1$
$y = 3x + 8$
$y = 4x - 10$