To determine the solution of two linear equations represented graphically, you can follow these steps:
Determining the Solution Graphically
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Graph the Equations: Start by graphing both linear equations on the same coordinate plane. Each equation in the form \( y = mx + b \) can be represented by plotting points that satisfy the equation and then drawing a straight line through those points. The slope \( m \) indicates the steepness of the line, and \( b \) is the y-intercept.
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Identify the Intersection Point: Once both lines are graphed, look for the point where the two lines intersect. The coordinates of this intersection point (let's say it is \((x, y)\)) represent the solution to the system of equations.
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Interpret the Intersection: The intersection point signifies that both equations are true for that set of \( x \) and \( y \) values, meaning it's the point where the two equations have the same output (y-value) for the same input (x-value).
Verifying the Solution
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Substitution Method: Take the coordinates of the intersection point \((x, y)\) and substitute them back into both original equations. If both equations hold true (i.e., both yield true statements), then the intersection point is indeed the solution.
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Use Algebraic Methods: You can also solve the system of equations algebraically using methods such as substitution or elimination. Substitute the value of \( y \) from one equation into the other, or align the equations for elimination, and solve the resulting equation for \( x \). Then use that \( x \) value to find \( y \).
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Check the Graph: If you have access to graphing technology (like a graphing calculator or software), you can input both equations and verify that the intersection point is calculated correctly.
Summary
By graphing the two linear equations, finding their intersection, and verifying it through substitution or algebraic methods, you can accurately determine and confirm the solution to the system of equations.
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