Solve the system of equation by graphing. Then classify the system. 3x + y = 4, 4x + 5y = -24. Solution:
Infinitity many solutions, no solution, System inconsistent/ consistent, Equation dependent/ independent.
To solve the system of equations by graphing, we'll start by graphing each equation on a coordinate plane.
1) Start with the equation 3x + y = 4.
- To graph this equation, rearrange it in the slope-intercept form y = mx + b, where m represents the slope and b represents the y-intercept.
- By subtracting 3x from both sides, we get: y = -3x + 4.
- This equation has a slope of -3 and a y-intercept of 4.
- Plot the y-intercept (0, 4) and use the slope to find additional points. For example, from the y-intercept, move one unit to the right and three units down to reach the point (1, 1).
2) Now let's graph the second equation, 4x + 5y = -24.
- Rearrange the equation in slope-intercept form.
- Subtract 4x from both sides: 5y = -4x - 24.
- Divide both sides by 5: y = -4/5x - 24/5.
- This equation has a slope of -4/5 and a y-intercept of -24/5.
- Plot the y-intercept (0, -24/5) and use the slope to find additional points.
Now, look at the graph of the two equations. If the lines intersect at a single point, it represents a unique solution, meaning there is one point of intersection. This occurs when the system is consistent and the equations are independent.
However, if the lines are parallel and never intersect, it indicates no solutions exist. In this case, the system would be inconsistent, and the equations would be independent.
Lastly, if the lines lie on top of each other and coincide for all points, it implies infinitely many solutions exist. This would indicate a consistent system with dependent equations.
By analyzing the graph, determine which of these scenarios applies to the system of equations 3x + y = 4 and 4x + 5y = -24.
To solve the system of equations by graphing, you will need to plot the graphs of both equations on the same coordinate plane and identify the point(s) where the two lines intersect, if any.
First, let's solve the system of equations:
1. 3x + y = 4
2. 4x + 5y = -24
To graph the first equation, let's rewrite it into slope-intercept form (y = mx + b):
1. y = -3x + 4
From this equation, you can determine that the y-intercept is 4 and the slope is -3 (change in y divided by change in x).
Now, let's graph the second equation:
2. 4x + 5y = -24
To convert this equation into slope-intercept form, we need to isolate y:
2. 5y = -4x - 24
y = (-4/5)x - 24/5
From this equation, you can determine that the y-intercept is -24/5 and the slope is -4/5.
Now that you have the equations in slope-intercept form, you can graph them on the same coordinate plane.
Choose some x-values, substitute them into the equations to find the corresponding y-values, and plot the points. Repeat this process at least twice for each equation to get multiple points on each line. Then, connect the points with a straight line.
After graphing both equations, analyze the intersection of the lines. If the lines intersect at a single point, that point represents the solution to the system. If the lines are parallel and do not intersect, there is no solution. If the lines overlap or coincide, there are infinitely many solutions.
By graphing the given system of equations, you can determine that the lines do not intersect at a single point but rather overlap. This indicates that the system has infinitely many solutions.
Therefore, the classification of this system is "infinitely many solutions."