Solve The Equation:${ 4x + 7y = -30 }$

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Introduction to Linear Equations

Linear equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. A linear equation is an equation in which the highest power of the variable(s) is 1. In this article, we will focus on solving a linear equation of the form $4x + 7y = -30$. We will use a step-by-step approach to solve this equation and provide a clear understanding of the concept.

Understanding the Equation

The given equation is $4x + 7y = -30$. This equation represents a linear relationship between two variables, x and y. The coefficients of x and y are 4 and 7, respectively, and the constant term is -30. To solve this equation, we need to isolate one of the variables.

Solving for x

To solve for x, we can use the following steps:

1. Isolate the term with x: We can start by isolating the term with x by subtracting 7y from both sides of the equation. This gives us $4x = -30 - 7y$.
2. Divide both sides by 4: To solve for x, we need to divide both sides of the equation by 4. This gives us $x = \frac{-30 - 7y}{4}$.

Solving for y

To solve for y, we can use the following steps:

1. Isolate the term with y: We can start by isolating the term with y by subtracting 4x from both sides of the equation. This gives us $7y = -30 - 4x$.
2. Divide both sides by 7: To solve for y, we need to divide both sides of the equation by 7. This gives us $y = \frac{-30 - 4x}{7}$.

Using Substitution Method

We can also use the substitution method to solve this equation. Let's say we want to solve for x. We can substitute the expression for y from the previous equation into the original equation. This gives us:

$4x + 7\left(\frac{-30 - 4x}{7}\right) = -30$

Simplifying this equation, we get:

$4x - 30 - 4x = -30$

This equation is true for all values of x, which means that x is a free variable. We can assign any value to x, and the equation will still be true.

Using Elimination Method

We can also use the elimination method to solve this equation. Let's say we want to solve for x. We can multiply the first equation by 7 and the second equation by 4. This gives us:

$28x + 49y = -210$ $16x + 28y = -120$

Subtracting the second equation from the first equation, we get:

$12x + 21y = -90$

Simplifying this equation, we get:

$4x + 7y = -30$

This is the same equation we started with, which means that we have not eliminated any variables.

Conclusion

In this article, we have discussed how to solve a linear equation of the form $4x + 7y = -30$. We have used a step-by-step approach to solve this equation and provided a clear understanding of the concept. We have also discussed two methods for solving linear equations: substitution method and elimination method. These methods can be used to solve linear equations with two variables.

Real-World Applications

Linear equations have many real-world applications. For example, in physics, linear equations are used to describe the motion of objects. In economics, linear equations are used to model the relationship between variables such as supply and demand. In engineering, linear equations are used to design and optimize systems.

Tips and Tricks

Here are some tips and tricks for solving linear equations:

• Use the substitution method: The substitution method is a powerful tool for solving linear equations. It involves substituting the expression for one variable into the original equation.
• Use the elimination method: The elimination method is another powerful tool for solving linear equations. It involves eliminating one variable by multiplying the equations by appropriate constants.
• Check your work: It's always a good idea to check your work by plugging the solution back into the original equation.
• Use a graphing calculator: A graphing calculator can be a useful tool for visualizing the solution to a linear equation.

Common Mistakes

Here are some common mistakes to avoid when solving linear equations:

• Not isolating the variable: Make sure to isolate the variable by subtracting or adding the appropriate terms.
• Not checking your work: Always check your work by plugging the solution back into the original equation.
• Not using the correct method: Make sure to use the correct method for solving the equation, such as substitution or elimination.
• Not simplifying the equation: Make sure to simplify the equation by combining like terms.

Final Thoughts

Solving linear equations is an important skill that has many real-world applications. By following the steps outlined in this article, you can solve linear equations with ease. Remember to use the substitution method and elimination method, and always check your work by plugging the solution back into the original equation. With practice and patience, you will become proficient in solving linear equations.

Q: What is a linear equation?

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

Q: How do I solve a linear equation?

A: To solve a linear equation, you can use the substitution method or the elimination method. The substitution method involves substituting the expression for one variable into the original equation, while the elimination method involves eliminating one variable by multiplying the equations by appropriate constants.

Q: What is the substitution method?

A: The substitution method is a technique used to solve linear equations. It involves substituting the expression for one variable into the original equation. For example, if we have the equation 2x + 3y = 5, we can substitute the expression for y from the equation 3y = 2x - 5 into the original equation.

Q: What is the elimination method?

A: The elimination method is a technique used to solve linear equations. It involves eliminating one variable by multiplying the equations by appropriate constants. For example, if we have the equations 2x + 3y = 5 and x - 2y = -3, we can multiply the second equation by 3 and add it to the first equation to eliminate the variable y.

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

A: To check your work when solving a linear equation, you can plug the solution back into the original equation. If the solution satisfies the equation, then it is correct.

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

A: Some common mistakes to avoid when solving linear equations include not isolating the variable, not checking your work, not using the correct method, and not simplifying the equation.

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

A: Yes, you can use a graphing calculator to solve linear equations. A graphing calculator can be a useful tool for visualizing the solution to a linear equation.

Q: How do I use a graphing calculator to solve a linear equation?

A: To use a graphing calculator to solve a linear equation, you can enter the equation into the calculator and use the "solve" function to find the solution.

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

A: Linear equations have many real-world applications, including physics, economics, and engineering. In physics, linear equations are used to describe the motion of objects. In economics, linear equations are used to model the relationship between variables such as supply and demand. In engineering, linear equations are used to design and optimize systems.

Q: Can I use linear equations to solve systems of equations?

A: Yes, you can use linear equations to solve systems of equations. A system of equations is a set of two or more linear equations that are solved simultaneously.

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

A: To use linear equations to solve a system of equations, you can use the substitution method or the elimination method. The substitution method involves substituting the expression for one variable into the other equation, while the elimination method involves eliminating one variable by multiplying the equations by appropriate constants.

Q: What are some tips for solving linear equations?

A: Some tips for solving linear equations include using the substitution method and elimination method, checking your work, and simplifying the equation. Additionally, it's a good idea to use a graphing calculator to visualize the solution to a linear equation.

Q: Can I use linear equations to solve quadratic equations?

A: No, you cannot use linear equations to solve quadratic equations. Quadratic equations are equations in which the highest power of the variable is 2, and they require a different set of techniques to solve.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula, which is x = (-b ± √(b^2 - 4ac)) / 2a. You can also use factoring or the quadratic formula to solve a quadratic equation.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include not using the correct formula, not simplifying the equation, and not checking your work.

Q: Can I use a graphing calculator to solve quadratic equations?

A: Yes, you can use a graphing calculator to solve quadratic equations. A graphing calculator can be a useful tool for visualizing the solution to a quadratic equation.

Q: How do I use a graphing calculator to solve a quadratic equation?

A: To use a graphing calculator to solve a quadratic equation, you can enter the equation into the calculator and use the "solve" function to find the solution.

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

A: Quadratic equations have many real-world applications, including physics, engineering, and economics. In physics, quadratic equations are used to describe the motion of objects. In engineering, quadratic equations are used to design and optimize systems. In economics, quadratic equations are used to model the relationship between variables such as supply and demand.