Ordered Pairs and Solutions

Oct 13, 2025 by leolavanderia.com.br 100 views

Solving Linear Equations: Determining Solutions to $5x + 3y = -4$

Introduction

In mathematics, solving linear equations is a fundamental concept that involves finding the values of variables that satisfy a given equation. In this article, we will focus on determining whether ordered pairs are solutions to the linear equation $5x + 3y = -4$ . This equation is a classic example of a linear equation in two variables, and solving it requires a clear understanding of the concept of solutions to linear equations.

What are Solutions to Linear Equations?

A solution to a linear equation is an ordered pair of values that makes the equation true. In other words, if we substitute the values of the ordered pair into the equation, the equation should hold true. For example, if we have the equation $2x + 3y = 5$ , then the ordered pair $(1, 2)$ is a solution to the equation because substituting $x = 1$ and $y = 2$ into the equation results in $2(1) + 3(2) = 2 + 6 = 8$ , which is not equal to 5. However, the ordered pair $(2, 1)$ is not a solution to the equation because substituting $x = 2$ and $y = 1$ into the equation results in $2(2) + 3(1) = 4 + 3 = 7$ , which is not equal to 5.

The Equation $5x + 3y = -4$

The equation $5x + 3y = -4$ is a linear equation in two variables, where the coefficients of the variables are $5$ and $3$ , and the constant term is $-4$ . To determine whether an ordered pair is a solution to this equation, we need to substitute the values of the ordered pair into the equation and check if the equation holds true.

Ordered Pairs and Solutions

Let's consider the ordered pairs given in the problem:

$(1, -3)$
$(-5, 7)$
$(4, 6)$
$(-6, -2)$

We need to determine whether each of these ordered pairs is a solution to the equation $5x + 3y = -4$ .

Checking the Ordered Pairs

To check whether an ordered pair is a solution to the equation, we need to substitute the values of the ordered pair into the equation and simplify.

Checking $(1, -3)$

Substituting $x = 1$ and $y = -3$ into the equation $5x + 3y = -4$ , we get:

$5(1) + 3(-3) = 5 - 9 = -4$

Since the equation holds true, the ordered pair $(1, -3)$ is a solution to the equation.

Checking $(-5, 7)$

Substituting $x = -5$ and $y = 7$ into the equation $5x + 3y = -4$ , we get:

$5(-5) + 3(7) = -25 + 21 = -4$

Since the equation holds true, the ordered pair $(-5, 7)$ is a solution to the equation.

Checking $(4, 6)$

Substituting $x = 4$ and $y = 6$ into the equation $5x + 3y = -4$ , we get:

$5(4) + 3(6) = 20 + 18 = 38$

Since the equation does not hold true, the ordered pair $(4, 6)$ is not a solution to the equation.

Checking $(-6, -2)$

Substituting $x = -6$ and $y = -2$ into the equation $5x + 3y = -4$ , we get:

$5(-6) + 3(-2) = -30 - 6 = -36$

Since the equation does not hold true, the ordered pair $(-6, -2)$ is not a solution to the equation.

Conclusion

In this article, we have determined whether each of the given ordered pairs is a solution to the linear equation $5x + 3y = -4$ . We have used the concept of solutions to linear equations to check whether each ordered pair satisfies the equation. The ordered pairs $(1, -3)$ and $(-5, 7)$ are solutions to the equation, while the ordered pairs $(4, 6)$ and $(-6, -2)$ are not solutions to the equation.

Discussion

The concept of solutions to linear equations is a fundamental concept in mathematics that has numerous applications in various fields, including science, engineering, and economics. Understanding how to determine whether an ordered pair is a solution to a linear equation is essential for solving problems in these fields. In this article, we have provided a clear and concise explanation of how to determine whether an ordered pair is a solution to the linear equation $5x + 3y = -4$ . We hope that this article has provided valuable insights and knowledge to readers who are interested in mathematics and problem-solving.

References

[1] "Linear Equations" by Khan Academy
[2] "Solutions to Linear Equations" by Math Open Reference
[3] "Linear Equations in Two Variables" by Purplemath

Introduction

In our previous article, we discussed how to determine whether an ordered pair is a solution to the linear equation $5x + 3y = -4$ . In this article, we will answer some of the most frequently asked questions about solutions to linear equations.

Q&A

Q: What is a solution to a linear equation?

A: A solution to a linear equation is an ordered pair of values that makes the equation true. In other words, if we substitute the values of the ordered pair into the equation, the equation should hold true.

Q: How do I determine whether an ordered pair is a solution to a linear equation?

A: To determine whether an ordered pair is a solution to a linear equation, we need to substitute the values of the ordered pair into the equation and simplify. If the equation holds true, then the ordered pair is a solution to the equation.

Q: What is the difference between a solution and a point on a graph?

A: A solution to a linear equation is an ordered pair of values that makes the equation true, while a point on a graph is a point that lies on the graph of the equation. Not all points on a graph are solutions to the equation, and not all solutions to the equation are points on the graph.

Q: Can a linear equation have multiple solutions?

A: Yes, a linear equation can have multiple solutions. For example, the equation $2x + 3y = 5$ has multiple solutions, including $(1, 2)$ and $(2, 1)$ .

Q: Can a linear equation have no solutions?

A: Yes, a linear equation can have no solutions. For example, the equation $2x + 3y = 5$ has no solution if we substitute $x = 1$ and $y = 2$ , because the equation becomes $2(1) + 3(2) = 2 + 6 = 8$ , which is not equal to 5.

Q: How do I find the solution to a linear equation?

A: To find the solution to a linear equation, we need to isolate one of the variables. We can do this by using algebraic manipulations, such as adding or subtracting the same value to both sides of the equation, or multiplying or dividing both sides of the equation by the same value.

Q: What is the significance of solutions to linear equations?

A: Solutions to linear equations are significant because they help us understand the behavior of the equation. For example, if we know that the equation $2x + 3y = 5$ has a solution, then we know that the graph of the equation will intersect the line $y = 2x - 5$ .

Conclusion

In this article, we have answered some of the most frequently asked questions about solutions to linear equations. We have discussed what a solution to a linear equation is, how to determine whether an ordered pair is a solution to a linear equation, and the significance of solutions to linear equations. We hope that this article has provided valuable insights and knowledge to readers who are interested in mathematics and problem-solving.

Discussion

The concept of solutions to linear equations is a fundamental concept in mathematics that has numerous applications in various fields, including science, engineering, and economics. Understanding how to determine whether an ordered pair is a solution to a linear equation is essential for solving problems in these fields. In this article, we have provided a clear and concise explanation of how to determine whether an ordered pair is a solution to a linear equation. We hope that this article has provided valuable insights and knowledge to readers who are interested in mathematics and problem-solving.

References

[1] "Linear Equations" by Khan Academy
[2] "Solutions to Linear Equations" by Math Open Reference
[3] "Linear Equations in Two Variables" by Purplemath