Introduction
In this blog article, we will be discussing the answer key for Unit 4 Linear Equations Homework 7. Linear equations are a fundamental concept in mathematics, and understanding how to solve them is crucial for success in algebra and beyond. This answer key will provide step-by-step solutions to the homework questions, helping you to check your work and ensure accuracy. So, let's dive in and explore the answers to Unit 4 Linear Equations Homework 7!
Question 1: Solving a Single Variable Equation
In this question, you are given a linear equation with a single variable and asked to solve for that variable. The equation may involve addition, subtraction, multiplication, or division operations. To solve this type of equation, you need to isolate the variable on one side of the equation and simplify the other side.
Step 1: Begin by simplifying both sides of the equation. Combine like terms and perform any necessary operations.
Step 2: Use inverse operations to isolate the variable on one side of the equation. If the variable is being added, subtract it from both sides. If the variable is being subtracted, add it to both sides. If the variable is being multiplied, divide both sides by the coefficient. If the variable is being divided, multiply both sides by the reciprocal.
Step 3: After isolating the variable, check your answer by substituting the value back into the original equation and ensuring that both sides are equal.
Let's apply these steps to the specific equation in question 1 and find the answer.
Question 2: Solving a System of Linear Equations
In this question, you are given a system of linear equations and asked to find the solution. A system of linear equations consists of two or more equations with the same variables. To solve this type of system, you can use the substitution method or the elimination method.
Substitution Method:
Step 1: Choose one equation and solve it for one variable in terms of the other variable.
Step 2: Substitute the expression obtained in step 1 into the other equation and solve for the remaining variable.
Step 3: Substitute the values of the variables obtained in step 2 back into either of the original equations and solve for the other variable.
Step 4: Check your solution by substituting the values of the variables into both equations and ensuring that both sides are equal.
Elimination Method:
Step 1: Multiply one or both equations by appropriate constants to create opposite coefficients for one of the variables.
Step 2: Add or subtract the equations to eliminate one variable.
Step 3: Solve the resulting equation for the remaining variable.
Step 4: Substitute the value of the variable obtained in step 3 back into either of the original equations and solve for the other variable.
Step 5: Check your solution by substituting the values of the variables into both equations and ensuring that both sides are equal.
Let's apply these methods to the specific system of equations in question 2 and find the solution.
Question 3: Graphing a Linear Equation
In this question, you are given a linear equation and asked to graph it on a coordinate plane. Graphing linear equations helps visualize the relationship between variables and enables us to identify solutions, intercepts, and other key information.
To graph a linear equation:
Step 1: Write the equation in slope-intercept form, y = mx + b, where m represents the slope and b represents the y-intercept.
Step 2: Identify the y-intercept by locating the point where the line crosses the y-axis. Plot this point on the coordinate plane.
Step 3: Use the slope to find additional points on the line. The slope represents the change in y divided by the change in x, so you can use this information to determine the coordinates of other points.
Step 4: Connect the points on the coordinate plane to create a straight line.
Let's apply these steps to the specific equation in question 3 and graph it.
Question 4: Writing an Equation from a Word Problem
In this question, you are given a word problem and asked to write a linear equation that represents the situation. Word problems provide real-world context for mathematical concepts and require us to translate verbal information into mathematical expressions.
To write an equation from a word problem:
Step 1: Identify the unknown variable. This is the quantity you are asked to find.
Step 2: Read the problem carefully and determine the relationship between the known quantities and the unknown variable.
Step 3: Write an equation that represents the relationship. This equation may involve addition, subtraction, multiplication, or division operations.
Step 4: Solve the equation to find the value of the unknown variable.
Let's apply these steps to the specific word problem in question 4 and write the corresponding equation.
Question 5: Solving a Literal Equation
In this question, you are given a literal equation and asked to solve for a specific variable. A literal equation contains two or more variables and requires us to isolate the desired variable.
To solve a literal equation:
Step 1: Identify the variable you want to solve for.
Step 2: Use inverse operations to isolate the variable on one side of the equation. Treat the other variables as constants.
Step 3: Simplify both sides of the equation and check your answer by substituting the value back into the original equation.
Let's apply these steps to the specific literal equation in question 5 and find the solution.
Question 6: Solving a Proportional Relationship
In this question, you are given a proportional relationship and asked to find the constant of proportionality. Proportional relationships involve a direct variation between two variables, meaning that as one variable increases or decreases, the other variable does as well.
To solve a proportional relationship:
Step 1: Write the relationship as an equation in the form y = kx, where k represents the constant of proportionality.
Step 2: Use the given data to find the value of the constant of proportionality.
Step 3: Check your answer by substituting the values back into the original equation and ensuring that both sides are equal.
Let's apply these steps to the specific proportional relationship in question 6 and find the constant of proportionality.
Question 7: Solving a Literal Proportion
In this question, you are given a literal proportion and asked to solve for a specific variable. A literal proportion contains two or more variables and requires us to find the value of the desired variable.
To solve a literal proportion:
Step 1: Cross-multiply the terms of the proportion to eliminate the fractions.
Step 2: Simplify both sides of the equation, treating the other variables as constants.
Step 3: Isolate the desired variable by using inverse operations.
Step 4: Simplify the equation and check your answer by substituting the value back into the original proportion.
Let's apply these steps to the specific literal proportion in question 7 and find the solution.
Question 8: Solving a System of Literal Equations
In this question, you are given a system of literal equations and asked to find the solution. A system of literal equations consists of two or more equations with two or more variables. To solve this type of system, you can use substitution or elimination methods, similar to solving a system of linear equations.
Substitution Method:
Step 1: Choose one equation and solve it for one variable in terms of the other variables.
Step 2: Substitute the expression obtained in step 1 into the other equations and solve for the remaining variables.
Step 3: Substitute the values of the variables obtained in step 2 back into any of the original equations and solve for the remaining variable.
Step 4: Check your solution by substituting the values of the variables into all of the original equations and ensuring that both sides are equal.
Elimination Method:
Step 1: Multiply one or both equations by appropriate constants to create opposite coefficients for one of the variables.
Step 2: Add or subtract the equations to eliminate one variable.
Step 3: Solve the resulting equation for the remaining variable.
Step 4: Substitute the value of the variable obtained in step 3 back into either of the original equations and solve for the other variables.
Step 5: Check your solution by substituting the values of the variables into all of the original equations and ensuring that both sides are equal.
Let's apply these methods to the specific system of literal equations in question 8 and find the solution.
Question 9: Solving a Linear Inequality
In this question, you are given