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Questions and Answers
What is the primary goal of solving linear equations?
What is the primary goal of solving linear equations?
What is the first step in the substitution method for solving linear equations?
What is the first step in the substitution method for solving linear equations?
How can you solve an equation in the form ax + by = c for x?
How can you solve an equation in the form ax + by = c for x?
What is the result of the substitution method?
What is the result of the substitution method?
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What is the purpose of substituting the resulting expression for that variable into the other equation?
What is the purpose of substituting the resulting expression for that variable into the other equation?
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How many steps are involved in the substitution method for solving linear equations?
How many steps are involved in the substitution method for solving linear equations?
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What is the first step in solving the system of linear equations using the substitution method?
What is the first step in solving the system of linear equations using the substitution method?
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What do we do after solving for one of the variables in the substitution method?
What do we do after solving for one of the variables in the substitution method?
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What is the form of the equation that can be solved for x or y by dividing both sides by a or b?
What is the form of the equation that can be solved for x or y by dividing both sides by a or b?
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What is the purpose of combining the terms in the substitution method?
What is the purpose of combining the terms in the substitution method?
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What is the final step in the substitution method?
What is the final step in the substitution method?
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What can be said about the substitution method?
What can be said about the substitution method?
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Study Notes
Solving Linear Equations: A Comprehensive Guide
In mathematics, linear equations are a fundamental part of algebra and are widely used to model various real-life situations. Solving linear equations involves finding the values of the variables that satisfy the given equation. In this article, we will discuss the methods for solving linear equations, focusing on the substitution method.
Substitution Method
The substitution method is an algebraic method used to solve linear equations. It involves solving one of the equations for one of the variables and then substituting the value obtained into the other equation. This process is repeated until a single equation with one variable is obtained, which can be easily solved.
Here's a step-by-step guide on how to solve a linear equation using the substitution method:
-
Identify the equation you want to solve for one variable. This is usually the equation that has fewer variables.
-
Solve the equation for one of the variables. If the equation is in the form
ax + by = c, whereaandbare constants, you can solve it forxby dividing both sides byaor foryby dividing both sides byb. -
Substitute the resulting expression for that variable into the other equation. Replace the variable in the second equation with the expression you obtained in step 2.
-
Solve the new equation for the remaining variable. If the equation is in the form
ax + by = c, whereaandbare constants, you can solve it forxby dividing both sides byaor foryby dividing both sides byb. -
Substitute the value of the remaining variable into one of the original equations to find the value of the other variable.
Example
Let's solve the system of linear equations:
2x + 3y = 9
x – y = 3
- We will solve the second equation for
x:
x – y = 3
x = y + 3
- Substitute
x = y + 3into the first equation:
2(y + 3) + 3y = 9
2y + 6 + 3y = 9
- Combine the terms:
5y + 6 = 9
5y = 9 - 6
- Solve for
y:
y = (9 - 6) / 5
y = 3 / 5
- Substitute
y = 3/5intox = y + 3:
x = (3/5) + 3
The solution for the system of linear equations is x = 3/5 + 3 and y = 3/5.
In conclusion, the substitution method is a simple and effective way to solve linear equations. It involves solving one of the equations for one of the variables and then substituting the value obtained into the other equation. By repeating this process, you can isolate the remaining variable and find the value of both variables.
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Description
Learn how to solve linear equations using the substitution method, a fundamental technique in algebra. This guide covers the step-by-step process of solving linear equations, including examples and illustrations.
