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Questions and Answers
What is the solution to the system of equations $x - y = 1$ and $2x + y = 7$?
What is the solution to the system of equations $x - y = 1$ and $2x + y = 7$?
By which method can a variable be eliminated in a system of linear equations?
By which method can a variable be eliminated in a system of linear equations?
If the system of equations is $2x - 3y = 6$ and $x + y = 4$, what is the solution?
If the system of equations is $2x - 3y = 6$ and $x + y = 4$, what is the solution?
What is the value of $x$ in the system $3x - 2y = 4$ and $2x + y = 6$?
What is the value of $x$ in the system $3x - 2y = 4$ and $2x + y = 6$?
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In the system of equations $4x + 2y = 10$ and $y - x = -3$, what is the solution?
In the system of equations $4x + 2y = 10$ and $y - x = -3$, what is the solution?
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What does the intersection of two linear equations on a coordinate plane represent?
What does the intersection of two linear equations on a coordinate plane represent?
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In graphing linear equations, what does 'm' represent in the slope-intercept form y = mx + b?
In graphing linear equations, what does 'm' represent in the slope-intercept form y = mx + b?
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What method involves solving one of the equations for a variable, then substituting it into another equation to simplify the system?
What method involves solving one of the equations for a variable, then substituting it into another equation to simplify the system?
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When graphing linear equations, what do the points of intersection between lines represent?
When graphing linear equations, what do the points of intersection between lines represent?
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What does the 'b' represent in the slope-intercept form y = mx + b?
What does the 'b' represent in the slope-intercept form y = mx + b?
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Study Notes
Solving Pairs of Linear Equations in Two Variables
When you encounter two linear expressions involving the same pair of variables (let's say (x) and (y)), their intersection usually represents a point where both equations hold true simultaneously. This section will delve into how we tackle pairs of such linear equations. We'll specifically explore graphical methods using coordinate planes, along with algebraic techniques like substitution and elimination.
Graphing Method
To visualize solutions, you can create a coordinate plane, plotting each equation by connecting points represented by its ordered pairs ((x, y)):
- Plot each line individually based on (y = mx + b) (the slope-intercept form), where (m) is the slope, and (b) is the (y-) intercept.
- Identify the points of intersection between lines when they cross. These locations represent all possible values of (x) and (y) satisfying both equations simultaneously.
Substitution Method
Given one of the equations, solve it for either variable — let's assume (x). Then substitute this expression into another equation, forming a single equation in only one remaining variable, typically (y). Solve that simplified equation, which yields exact or approximate values for (y). Finally, plug these back into the original (x=)formula from step 1 to find corresponding values for (x).
For example, consider the following system: [ \begin{aligned} & x - y = 1 \ & 2x + y = 7 \end{aligned} ]
First, rewrite the first equation as (x=y+1.) Now substitute this into the second equation: (2(y + 1) + y = 7,) which becomes (3y = 4;) therefore, (y = \frac{4}{3}.) Next, plug this value of (y) back into our initial replacement formula ((x=y+1),) yielding (x =\frac{4}{3}+1 = \frac{7}{3}). Thus, the solution is (\left(\dfrac{7}{3}, \dfrac{4}{3}\right)).
Elimination Method
Another approach involves manipulating the coefficients so that one variable disappears entirely, leaving behind only one equation in terms of the other. For instance, if you multiply the first equation by a certain constant and then add it to the second equation, the coefficient of some variable may cancel out—leaving us with a single equation in only one unknown.
Suppose we have the same system of equations mentioned above: [ \begin{aligned} & x - y = 1 \ & 2x + y = 7 \end{aligned} ]
Multiplying the first equation by 2 gives us (2x - 2y = 2.) Adding this to the second equation results in (3x = 9;$ thus, $x = 3.$ Subsequently, finding (y) via the first equation (since (x) has already been solved): (y = x - 1 = 2.) So, once again, the unique solution is (\left(3, 2\right)).
In conclusion, there exist multiple approaches to solving systems of linear equations, including graphical analysis alongside more traditional algebraic strategies. Each technique offers unique insights and practical applications depending on the specific problem at hand.
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Description
Learn how to solve pairs of linear equations in two variables using graphical, substitution, and elimination methods. Explore techniques such as plotting equations on coordinate planes, substituting variables to simplify equations, and manipulating coefficients to eliminate variables. Practice solving systems of linear equations and finding unique solutions.
