Complete Each Ordered Pair So That It Is A Solution Of The Given Linear Equation.$\[ X - 2y = -3 \\]1. For The Ordered Pair \[$(\square, -4)\$\], Solve For \[$x\$\].2. For The Ordered Pair \[$(3, \square)\$\], Solve For

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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 completing ordered pairs for given linear equations. We will explore two different scenarios: solving for x when the ordered pair is given, and solving for y when the ordered pair is given.

Solving for x when the ordered pair is given

When we are given an ordered pair and a linear equation, we can use the equation to solve for the value of x. Let's consider the following equation:

x−2y=−3{ x - 2y = -3 }

We are given the ordered pair (□,−4)${(\square, -4)\$}. To solve for x, we can substitute the value of y into the equation and then isolate x.

Step 1: Substitute the value of y into the equation

We are given that y = -4. Substituting this value into the equation, we get:

x−2(−4)=−3{ x - 2(-4) = -3 }

Step 2: Simplify the equation

Simplifying the equation, we get:

x+8=−3{ x + 8 = -3 }

Step 3: Isolate x

To isolate x, we can subtract 8 from both sides of the equation:

x=−3−8{ x = -3 - 8 }

x=−11{ x = -11 }

Therefore, the value of x is -11.

Solving for y when the ordered pair is given

When we are given an ordered pair and a linear equation, we can use the equation to solve for the value of y. Let's consider the following equation:

x−2y=−3{ x - 2y = -3 }

We are given the ordered pair (3,â–¡)${(3, \square)\$}. To solve for y, we can substitute the value of x into the equation and then isolate y.

Step 1: Substitute the value of x into the equation

We are given that x = 3. Substituting this value into the equation, we get:

3−2y=−3{ 3 - 2y = -3 }

Step 2: Simplify the equation

Simplifying the equation, we get:

−2y=−6{ -2y = -6 }

Step 3: Isolate y

To isolate y, we can divide both sides of the equation by -2:

y=−6−2{ y = \frac{-6}{-2} }

y=3{ y = 3 }

Therefore, the value of y is 3.

Conclusion

In this article, we have explored how to complete ordered pairs for given linear equations. We have seen how to solve for x when the ordered pair is given, and how to solve for y when the ordered pair is given. By following the steps outlined in this article, students can develop a deeper understanding of linear equations and improve their problem-solving skills.

Examples

Here are some examples of completing ordered pairs for given linear equations:

Example 1

Given the equation x−2y=−3{ x - 2y = -3 } and the ordered pair (□,−4)${(\square, -4)\$}, solve for x.

x−2(−4)=−3{ x - 2(-4) = -3 }

x+8=−3{ x + 8 = -3 }

x=−3−8{ x = -3 - 8 }

x=−11{ x = -11 }

Example 2

Given the equation x−2y=−3{ x - 2y = -3 } and the ordered pair (3,□)${(3, \square)\$}, solve for y.

3−2y=−3{ 3 - 2y = -3 }

−2y=−6{ -2y = -6 }

y=−6−2{ y = \frac{-6}{-2} }

y=3{ y = 3 }

Practice Problems

Here are some practice problems to help students develop their skills in completing ordered pairs for given linear equations:

Problem 1

Given the equation x−2y=−3{ x - 2y = -3 } and the ordered pair (□,−2)${(\square, -2)\$}, solve for x.

Problem 2

Given the equation x−2y=−3{ x - 2y = -3 } and the ordered pair (4,□)${(4, \square)\$}, solve for y.

Problem 3

Given the equation x−2y=−3{ x - 2y = -3 } and the ordered pair (□,1)${(\square, 1)\$}, solve for x.

Problem 4

Given the equation x−2y=−3{ x - 2y = -3 } and the ordered pair (2,□)${(2, \square)\$}, solve for y.

Answer Key

Here is the answer key for the practice problems:

Problem 1

x=−1{ x = -1 }

Problem 2

y=−52{ y = -\frac{5}{2} }

Problem 3

x=−5{ x = -5 }

Problem 4

y=−12{ y = -\frac{1}{2} }

Q: What is an ordered pair?

A: An ordered pair is a pair of numbers that are written in a specific order, usually in the form (x, y). For example, (3, 4) is an ordered pair where x = 3 and y = 4.

Q: What is a linear equation?

A: A linear equation is an equation that can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables. For example, 2x + 3y = 5 is a linear equation.

Q: How do I complete an ordered pair for a linear equation?

A: To complete an ordered pair for a linear equation, you need to substitute the values of x and y into the equation and then solve for the other variable. For example, if you have the equation x - 2y = -3 and the ordered pair (x, y) = (3, -4), you can substitute x = 3 and y = -4 into the equation and solve for the other variable.

Q: What if I have a linear equation with two variables and I want to find the value of one variable?

A: If you have a linear equation with two variables and you want to find the value of one variable, you can use the equation to solve for the other variable. For example, if you have the equation x - 2y = -3 and you want to find the value of y, you can substitute x = 3 into the equation and solve for y.

Q: Can I use the same method to solve for x and y in a linear equation?

A: Yes, you can use the same method to solve for x and y in a linear equation. The only difference is that you need to substitute the values of x and y into the equation and then solve for the other variable.

Q: What if I have a linear equation with a fraction?

A: If you have a linear equation with a fraction, you can use the same method to solve for x and y. However, you need to be careful when simplifying the equation and make sure that you are not multiplying or dividing by zero.

Q: Can I use a calculator to solve for x and y in a linear equation?

A: Yes, you can use a calculator to solve for x and y in a linear equation. However, it's always a good idea to check your work by plugging the values back into the equation to make sure that they are correct.

Q: What if I have a system of linear equations?

A: If you have a system of linear equations, you can use the same method to solve for x and y. However, you need to be careful when solving the system of equations and make sure that you are not multiplying or dividing by zero.

Q: Can I use a graphing calculator to solve a system of linear equations?

A: Yes, you can use a graphing calculator to solve a system of linear equations. However, it's always a good idea to check your work by plugging the values back into the equation to make sure that they are correct.

Q: What if I have a linear equation with a negative coefficient?

A: If you have a linear equation with a negative coefficient, you can use the same method to solve for x and y. However, you need to be careful when simplifying the equation and make sure that you are not multiplying or dividing by zero.

Q: Can I use a linear equation to model real-world problems?

A: Yes, you can use a linear equation to model real-world problems. For example, you can use a linear equation to model the cost of a product based on the number of units sold.

Q: What are some common applications of linear equations?

A: Some common applications of linear equations include:

  • Modeling the cost of a product based on the number of units sold
  • Modeling the distance traveled by an object based on the time it takes to travel
  • Modeling the amount of money in a bank account based on the interest rate and the time period
  • Modeling the population of a city based on the growth rate and the initial population

Q: Can I use linear equations to solve problems in other areas of mathematics?

A: Yes, you can use linear equations to solve problems in other areas of mathematics, such as algebra, geometry, and trigonometry.