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Questions and Answers
In the substitution method, if you have two equations, $3x + y = \frac{5}{3}$ and $y = 2x + \frac{1}{3}$, what is the first step after substituting the second equation into the first?
In the substitution method, if you have two equations, $3x + y = \frac{5}{3}$ and $y = 2x + \frac{1}{3}$, what is the first step after substituting the second equation into the first?
- Isolate $x$ by moving all $x$ terms to one side and constants to the other.
- Solve for $y$ in terms of $x$ in the resulting equation.
- Directly solve for a numerical value of $y$.
- Simplify the equation by combining like terms containing $x$. (correct)
Given the equations $3x + y = \frac{5}{3}$ and $y = 2x + \frac{1}{3}$, if you solve for $x$ and find that $x = 1$, what is the next step to find the value of $y$?
Given the equations $3x + y = \frac{5}{3}$ and $y = 2x + \frac{1}{3}$, if you solve for $x$ and find that $x = 1$, what is the next step to find the value of $y$?
- Use the additive inverse property to isolate $y$.
- Set up a new equation to directly solve for $y$.
- Substitute $x = 1$ into either of the original equations to solve for $y$. (correct)
- Average the two equations to eliminate $y$
If the solution to a system of equations is the intersection point (1,$\frac{7}{3}$), what does this imply about these values when substituted into the original equations?
If the solution to a system of equations is the intersection point (1,$\frac{7}{3}$), what does this imply about these values when substituted into the original equations?
- The values satisfy both equations simultaneously. (correct)
- The values do not satisfy any of the equations.
- The values satisfy only one of the equations.
- The values satisfy both equations, but only if they are first manipulated algebraically.
Using the addition method, given two equations $2x + 3y = 7$ and $x - 3y = 5$, what is the result of adding the two equations together?
Using the addition method, given two equations $2x + 3y = 7$ and $x - 3y = 5$, what is the result of adding the two equations together?
In the addition method, one step involves ensuring that when the equations are added, one of the variables is eliminated. What condition must be true for the coefficients of one variable to ensure it gets eliminated?
In the addition method, one step involves ensuring that when the equations are added, one of the variables is eliminated. What condition must be true for the coefficients of one variable to ensure it gets eliminated?
Consider two linear equations. After correctly applying either the substitution or addition method, you arrive at an equation where all variables are eliminated, resulting in a false statement (e.g., $0 = 12$). What does this indicate about the system of equations?
Consider two linear equations. After correctly applying either the substitution or addition method, you arrive at an equation where all variables are eliminated, resulting in a false statement (e.g., $0 = 12$). What does this indicate about the system of equations?
Why is it important to check the solution obtained after solving a system of equations using either the substitution or addition method?
Why is it important to check the solution obtained after solving a system of equations using either the substitution or addition method?
What is the purpose of solving a system of two linear equations?
What is the purpose of solving a system of two linear equations?
Given the system of equations $2x + 3y = 7$ and $x - 3y = 5$, what is the value of $x$ at the point of intersection?
Given the system of equations $2x + 3y = 7$ and $x - 3y = 5$, what is the value of $x$ at the point of intersection?
Given the system of equations $2x + 3y = 7$ and $x - 3y = 5$, and knowing that $x = 4$ at the point of intersection, what is the value of $y$?
Given the system of equations $2x + 3y = 7$ and $x - 3y = 5$, and knowing that $x = 4$ at the point of intersection, what is the value of $y$?
Flashcards
Einsetzungsverfahren (Substitution Method)
Einsetzungsverfahren (Substitution Method)
A method to solve systems of equations by substituting one equation into another.
Schnittpunkt (Intersection Point)
Schnittpunkt (Intersection Point)
The point where two lines intersect on a graph.
Additionsverfahren (Addition Method)
Additionsverfahren (Addition Method)
A method to solve systems of equations by adding the equations to eliminate one variable.
Ersetzen (Substitution)
Ersetzen (Substitution)
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Lösen (Solve)
Lösen (Solve)
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Study Notes
- Study notes on equation solving.
Substitution Method
- Solve:
- Equation 1: 3x + y = 5/3
- Equation 2: y = 2x + 1/3
- Due to Equation 2, a y- value exists
- Solution: y = 2x + 1/3
- In Equation 1, y can be replaced with 2x + 1/3
- 3x + 2x + 1/3 = 5/3
- Equation can focus on 5x = 15/3
- Therefore x = 1.
- If x is inserted into one of the equations, so y = 1/3
- Intersection point: (1 | 1/3)
Addition Method
- Solve:
- Equation 1: 2x + 3y = 7
- Equation 2: x - 3y = 5
- The sum of both terms on the left side has to equal the sum on the right side.
- Adding on both sides leads to an equation where y is not present.
- Equation 1 + Equation 2 = 2x + 3y + x - 3y = 7 + 5
- 3x + 0y = 12
- x = 4
- Substituting x into an equation: y = -1
- Intersection point: (4 | -1)
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