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Questions and Answers
What is the solution to the system of equations represented by the point (4, 1)?
What is the solution to the system of equations represented by the point (4, 1)?
When graphing linear equations, what is the significance of finding intercepts?
When graphing linear equations, what is the significance of finding intercepts?
For the equation y = 2x + 1, what would be the y-intercept?
For the equation y = 2x + 1, what would be the y-intercept?
How many points are needed to uniquely define a straight line graph?
How many points are needed to uniquely define a straight line graph?
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What does solving systems of linear equations by substitution enable us to do?
What does solving systems of linear equations by substitution enable us to do?
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In graphing linear equations, what does the slope of a line represent?
In graphing linear equations, what does the slope of a line represent?
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If a system of linear equations is being solved using the substitution method, what is the first step?
If a system of linear equations is being solved using the substitution method, what is the first step?
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In the given example of linear equations, what is the next step after substituting y's expression into the second equation?
In the given example of linear equations, what is the next step after substituting y's expression into the second equation?
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What is the purpose of moving everything to one side in the equation for solving x?
What is the purpose of moving everything to one side in the equation for solving x?
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What is the significance of dividing both sides by -7/2 during the solution process?
What is the significance of dividing both sides by -7/2 during the solution process?
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Which part of solving a system of linear equations using substitution involves expressing one variable in terms of the other?
Which part of solving a system of linear equations using substitution involves expressing one variable in terms of the other?
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What does distributing the 2 in step 4 achieve in solving linear equations?
What does distributing the 2 in step 4 achieve in solving linear equations?
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Study Notes
Linear Equations with Two Variables
When it comes to understanding and solving mathematical problems involving quantities that change in a straight line, we encounter linear equations with two variables. These equations take the form ax + by = c, where (a) and (b) are the coefficients of the variables (x) and (y), and (c) is the constant term. Let's delve into two common methods for solving such equations and graphing their solutions.
Solving by Substitution
One way to tackle linear equations with two variables is using substitution. This method involves solving for one variable in one equation and then substituting that expression into the second equation.
Example: Solve the following system of linear equations using substitution:
[ 3x + 2y = 5 ]
[ x + 3y = 7 ]
- First, find an expression for (y) in terms of (x) using one of the equations, say the first one:
[ y = \frac{5 - 3x}{2} ]
- Substitute this expression for (y) into the second equation:
[ x + 3 \left( \frac{5 - 3x}{2} \right) = 7 ]
- Simplify the equation:
[ x + \frac{15 - 9x}{2} = 7 ]
- Distribute the 2:
[ x + \frac{15}{2} - \frac{9}{2}x = 7 ]
- Combine the terms with (x):
[ - \frac{7}{2}x + \frac{15}{2} = 7 ]
- Move everything to one side to set up the equation for solving for (x):
[ - \frac{7}{2}x + \frac{35}{2} = 7 ]
- Move the constant term to the right side:
[ - \frac{7}{2}x = - \frac{28}{2} ]
- Divide both sides by -(\frac{7}{2}):
[ x = \frac{-28}{-7} = 4 ]
- Substitute the value of (x) back into the expression for (y):
[ y = \frac{5 - 3(4)}{2} = 1 ]
- The solution is ((4, 1)).
Graphing Linear Equations
Plotting linear equations in two-dimensional space is a great way to visualize the relationship between the variables. The graph of a linear equation in two variables is a straight line.
Steps for graphing linear equations:
- Find an intercept of the equation by setting one of the variables to zero.
- Plot the intercepts.
- Choose at least one more point from the equation.
- Draw a straight line through the two points.
Example: Plot the equation (y = 2x + 1).
- To find the (x)-intercept, set (y) to zero:
[ 0 = 2x + 1 ]
- Solve for (x):
[ x = - \frac{1}{2} ]
- To find the (y)-intercept, set (x) to zero:
[ y = 2(0) + 1 = 1 ]
- Plot the points ((- \frac{1}{2}, 1)) and ((0, 1)).
- Choose another point from the equation, for example, ((1, 3)).
- Draw a line through the points ((- \frac{1}{2}, 1)) and ((1, 3)).
The graph of (y = 2x + 1) is a straight line passing through the points ((- \frac{1}{2}, 1)) and ((1, 3)).
Graphing and solving linear equations with two variables are invaluable tools for understanding and modeling real-world scenarios. The ability to solve systems of linear equations by substitution, as well as to graph linear equations, opens the door to a wide range of applications in fields such as economics, finance, and engineering.
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Description
Dive into the realm of linear equations with two variables, exploring methods such as solving by substitution and graphing. Learn how to solve systems of equations using substitution and visually represent linear equations on a graph. Enhance your understanding of two-variable linear equations for practical applications in various fields.
