Answer:
Step-by-step explanation:
The given system of equations is:
y = 13x - 8 (Equation 1)
y = -53x - 2 (Equation 2)
To find the solution to the system of equations, we need to determine the values of x and y that satisfy both equations simultaneously.
Solving the System of Equations:
To solve the system of equations, we can set the two equations equal to each other:
13x - 8 = -53x - 2
Now, we can solve for x:
13x + 53x = -2 + 8
66x = 6
x = 6/66
x = 1/11
Substituting the value of x back into either of the original equations, we can find the value of y:
y = 13(1/11) - 8
y = 13/11 - 88/11
y = -75/11
Therefore, the solution to the system of equations is x = 1/11 and y = -75/11.
Graphical Representation:
To understand how the graph shows this solution, we can plot the two equations on a graph.
The graph of Equation 1, y = 13x - 8, is a straight line with a slope of 13 and a y-intercept of -8. It will pass through the point (0, -8) and have a positive slope, meaning it will rise as x increases.
The graph of Equation 2, y = -53x - 2, is also a straight line with a slope of -53 and a y-intercept of -2. It will pass through the point (0, -2) and have a negative slope, meaning it will fall as x increases.
By plotting these two lines on the same graph, we can visually determine the point of intersection, which represents the solution to the system of equations. In this case, the point of intersection is approximately (1/11, -75/11), which matches the solution we obtained algebraically.
The graph visually confirms that the coordinates (1/11, -75/11) satisfy both Equation 1 and Equation 2, making it the solution to the system of equations.
Conclusion:
The solution to the system of equations y = 13x - 8 and y = -53x - 2 is x = 1/11 and y = -75/11. The graph of the equations shows that the lines intersect at the point (1/11, -75/11), confirming this solution.
