Which graph represents the solution to the given system?
-2x+5y=-10 and -3x+5y=-20
The two equations can be rewritten in slope-intercept form as:
y = (2/5)x - 2 and y = (3/5)x - 4
By comparing the slopes (2/5 and 3/5), we can see that the lines are not parallel.
To find the solution, we need to find the point where the two lines intersect.
By graphing the equations, we find that the lines intersect at the point (10, 2).
Therefore, the graph of the solution to the given system is a single point at (10, 2), represented by graph D.
To determine which graph represents the solution to the system of equations, we need to solve the system first.
Let's solve the system of equations step-by-step using the method of substitution:
STEP 1: Solve one equation for one variable in terms of the other variable.
-2x + 5y = -10 --> -2x = -10 - 5y --> x = (10 + 5y) / 2
STEP 2: Substitute the expression for the variable obtained in step 1 into the other equation, and solve for the remaining variable.
-3x + 5y = -20
-3((10 + 5y) / 2) + 5y = -20
-30 - 15y + 10y = -40
-5y = -10
y = -10 / -5
y = 2
STEP 3: Substitute the value of y back into the expression for x obtained in step 1, and solve for x.
x = (10 + 5(2)) / 2
x = (10 + 10) / 2
x = 20 / 2
x = 10
Therefore, the solution to the system of equations -2x + 5y = -10 and -3x + 5y = -20 is x = 10, y = 2.
Now, let's check which graph represents this solution. Unfortunately, since we do not have any graphs provided, it is not possible to determine the specific graph that represents the solution visually. However, to verify the solution, the values x = 10 and y = 2 should satisfy both equations when substituted into them.
To find the solution to a system of equations, we need to graph the two equations and find the intersection point of the two lines. Since we have two equations with two variables, the solution to the system is the coordinates of the intersection point.
Let's solve the system step by step:
1. Let's rearrange the first equation -2x + 5y = -10 to solve for y:
-2x + 5y = -10
5y = 2x - 10
y = (2/5)x - 2
2. Now, let's rearrange the second equation -3x + 5y = -20 to solve for y:
-3x + 5y = -20
5y = 3x - 20
y = (3/5)x - 4
3. We now have both equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
Now, let's graph the two equations on the same graph:
- Plot the points for the first equation: y = (2/5)x - 2
Choose x values and calculate corresponding y values:
Let's choose x = 0:
y = (2/5)(0) - 2
y = -2
(0, -2) is one point on the first equation.
Let's choose x = 5:
y = (2/5)(5) - 2
y = 2 - 2
y = 0
(5, 0) is another point on the first equation.
- Plot the points for the second equation: y = (3/5)x - 4
Similarly, choose x values and calculate corresponding y values:
Let's choose x = 0:
y = (3/5)(0) - 4
y = -4
(0, -4) is one point on the second equation.
Let's choose x = 5:
y = (3/5)(5) - 4
y = 3 - 4
y = -1
(5, -1) is another point on the second equation.
- Plot the points (0, -2), (5, 0), (0, -4), and (5, -1) on the same graph.
- Draw a straight line passing through points (0, -2) and (5, 0). This line represents the first equation.
- Draw a straight line passing through points (0, -4) and (5, -1). This line represents the second equation.
The point where the two lines intersect is the solution to the system of equations.
If the lines do not intersect, it means that the system has no solution.
Based on the graph, we can determine the solution to the given system of equations.