Given F ( X ) = 5 − 3 X F(x)=5-3x F ( X ) = 5 − 3 X , If F ( X ) = − 19 F(x)=-19 F ( X ) = − 19 , Find X X X .Given G ( X ) = 2 X + 9 G(x)=2x+9 G ( X ) = 2 X + 9 , If G ( X ) = 15 G(x)=15 G ( X ) = 15 , Find X X X .Given H ( X ) = − X − 14 H(x)=-x-14 H ( X ) = − X − 14 , If H ( X ) = − 2 H(x)=-2 H ( X ) = − 2 , Find X X X .
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 explore three different linear equations and provide step-by-step solutions to find the value of x. We will use the given functions f(x), g(x), and h(x) to solve for x when the function is equal to a specific value.
Equation 1: Solving f(x) = -19
Given the function f(x) = 5 - 3x, we need to find the value of x when f(x) = -19.
Step 1: Write down the equation
f(x) = 5 - 3x -19 = 5 - 3x
Step 2: Isolate the variable x
To isolate x, we need to get rid of the constant term on the right-hand side. We can do this by subtracting 5 from both sides of the equation.
-19 - 5 = -3x -24 = -3x
Step 3: Solve for x
Now that we have -24 on the left-hand side, we can divide both sides of the equation by -3 to solve for x.
x = -24 / -3 x = 8
Therefore, when f(x) = -19, the value of x is 8.
Equation 2: Solving g(x) = 15
Given the function g(x) = 2x + 9, we need to find the value of x when g(x) = 15.
Step 1: Write down the equation
g(x) = 2x + 9 15 = 2x + 9
Step 2: Isolate the variable x
To isolate x, we need to get rid of the constant term on the right-hand side. We can do this by subtracting 9 from both sides of the equation.
15 - 9 = 2x 6 = 2x
Step 3: Solve for x
Now that we have 6 on the left-hand side, we can divide both sides of the equation by 2 to solve for x.
x = 6 / 2 x = 3
Therefore, when g(x) = 15, the value of x is 3.
Equation 3: Solving h(x) = -2
Given the function h(x) = -x - 14, we need to find the value of x when h(x) = -2.
Step 1: Write down the equation
h(x) = -x - 14 -2 = -x - 14
Step 2: Isolate the variable x
To isolate x, we need to get rid of the constant term on the right-hand side. We can do this by adding 14 to both sides of the equation.
-2 + 14 = -x 12 = -x
Step 3: Solve for x
Now that we have 12 on the left-hand side, we can multiply both sides of the equation by -1 to solve for x.
x = -12
Therefore, when h(x) = -2, the value of x is -12.
Conclusion
In this article, we have solved three different linear equations using the given functions f(x), g(x), and h(x). We have shown that by following the steps of isolating the variable x and solving for x, we can find the value of x when the function is equal to a specific value. These skills are essential for students to master in order to solve linear equations and apply them to real-world problems.
Key Takeaways
- To solve a linear equation, we need to isolate the variable x and solve for x.
- We can isolate x by getting rid of the constant term on the right-hand side of the equation.
- We can solve for x by dividing or multiplying both sides of the equation by a constant.
- Linear equations are a fundamental concept in mathematics and are used to model real-world problems.
Practice Problems
- Given the function f(x) = 2x + 5, find the value of x when f(x) = 11.
- Given the function g(x) = x - 3, find the value of x when g(x) = 7.
- Given the function h(x) = -2x + 1, find the value of x when h(x) = -5.
Answer Key
- x = 3
- x = 10
- x = 3
Solving Linear Equations: A Q&A Guide =====================================
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 provide a Q&A guide to help students understand and solve linear equations.
Q1: What is a linear equation?
A linear equation is an equation in which the highest power of the variable (x) is 1. It can be written in the form ax + b = c, where a, b, and c are constants.
Q2: How do I solve a linear equation?
To solve a linear equation, you need to isolate the variable x and solve for x. You can do this by getting rid of the constant term on the right-hand side of the equation and then solving for x.
Q3: What is the first step in solving a linear equation?
The first step in solving a linear equation is to write down the equation and identify the variable x. Then, you need to isolate x by getting rid of the constant term on the right-hand side of the equation.
Q4: How do I isolate x in a linear equation?
To isolate x in a linear equation, you can add or subtract the same value to both sides of the equation. You can also multiply or divide both sides of the equation by a constant to isolate x.
Q5: 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 (x) is 1, while a quadratic equation is an equation in which the highest power of the variable (x) is 2.
Q6: Can I use algebraic methods to solve linear equations?
Yes, you can use algebraic methods to solve linear equations. Some common algebraic methods include adding or subtracting the same value to both sides of the equation, multiplying or dividing both sides of the equation by a constant, and using inverse operations to isolate x.
Q7: How do I check my solution to a linear equation?
To check your solution to a linear equation, you need to plug the value of x back into the original equation and see if it is true. If the equation is true, then your solution is correct.
Q8: What are some common mistakes to avoid when solving linear equations?
Some common mistakes to avoid when solving linear equations include:
- Not isolating x correctly
- Not checking the solution
- Not using inverse operations correctly
- Not simplifying the equation
Q9: Can I use technology to solve linear equations?
Yes, you can use technology to solve linear equations. Some common tools include calculators, computer software, and online resources.
Q10: How do I apply linear equations to real-world problems?
Linear equations can be used to model real-world problems in a variety of fields, including physics, engineering, economics, and more. Some common applications of linear equations include:
- Modeling population growth
- Calculating interest rates
- Determining the cost of goods
- Solving problems in physics and engineering
Conclusion
In this article, we have provided a Q&A guide to help students understand and solve linear equations. We have covered topics such as the definition of a linear equation, how to solve a linear equation, and common mistakes to avoid. We have also discussed how to apply linear equations to real-world problems and how to use technology to solve linear equations.
Key Takeaways
- A linear equation is an equation in which the highest power of the variable (x) is 1.
- To solve a linear equation, you need to isolate the variable x and solve for x.
- You can use algebraic methods, technology, and real-world applications to solve linear equations.
- Common mistakes to avoid when solving linear equations include not isolating x correctly, not checking the solution, and not using inverse operations correctly.
Practice Problems
- Given the function f(x) = 2x + 5, find the value of x when f(x) = 11.
- Given the function g(x) = x - 3, find the value of x when g(x) = 7.
- Given the function h(x) = -2x + 1, find the value of x when h(x) = -5.
Answer Key
- x = 3
- x = 10
- x = 3