Review the graphs of a system of two linear equations in two variables: x−5y=7 and 3x−2y=−4 . Find the solution to both equations. (1 point) The intersection point is ().
To find the solution to the system of equations, we need to find the values of x and y that satisfy both equations.
The first equation is x - 5y = 7, which we can rewrite as x = 5y + 7.
Substituting this expression for x into the second equation, we get:
3(5y + 7) - 2y = -4
15y + 21 - 2y = -4
13y + 21 = -4
13y = -25
y = -25/13
Now, substitute this value of y back into the first equation to find x:
x = 5(-25/13) + 7
x = -125/13 + 7
x = -125/13 + 91/13
x = -34/13
So the solution to the system of equations is (x, y) = (-34/13, -25/13).
To find the solution to the system of equations, let's first review the graphs of the given equations:
Equation 1: x - 5y = 7
Equation 2: 3x - 2y = -4
To graph each equation, we can rearrange them into slope-intercept form (y = mx + b):
Equation 1: y = (1/5)x - 7/5
Equation 2: y = (3/2)x + 2
Now we can plot the graphs:
Equation 1:
- Start by plotting the y-intercept, which is -7/5. This is the point (0, -7/5).
- Then, use the slope of 1/5 to find another point. Since the slope is positive, move up 1 unit and right 5 units from the y-intercept. This gives us the point (5, -6/5).
- Draw a line through these two points to represent Equation 1.
Equation 2:
- Start by plotting the y-intercept, which is 2. This is the point (0, 2).
- Then, use the slope of 3/2 to find another point. Since the slope is positive, move up 3 units and right 2 units from the y-intercept. This gives us the point (2, 5).
- Draw a line through these two points to represent Equation 2.
The graph of Equation 1 is a downward-sloping line, while the graph of Equation 2 is an upward-sloping line.
To find the solution, we look for the point at which the two lines intersect. From the graph, we can see that the intersection point is approximately (4, -1).
Therefore, the solution to the system of equations is (4, -1).
To find the solution to a system of linear equations, you can use either the substitution method or the elimination method. In this case, we'll use the elimination method.
Step 1: Begin by rearranging each equation in the standard form of Ax + By = C.
Equation 1: x - 5y = 7
Equation 2: 3x - 2y = -4
Step 2: Multiply each equation by appropriate coefficients to eliminate one variable.
To eliminate the x variable, we'll multiply Equation 1 by 3 and Equation 2 by 1.
3(x - 5y) = 3(7) [Equation 1 multiplied by 3]
1(3x - 2y) = 1(-4) [Equation 2 multiplied by 1]
Which simplifies to:
3x - 15y = 21
3x - 2y = -4
Step 3: Subtract one equation from the other to eliminate the x variable.
(3x - 15y) - (3x - 2y) = 21 - (-4)
3x - 15y - 3x + 2y = 21 + 4
-13y = 25
Step 4: Solve for y.
Divide both sides of the equation by -13:
y = -25/13
Step 5: Substitute the value of y into one of the original equations to solve for x.
Using Equation 1: x - 5(-25/13) = 7
x + 125/13 = 7
x = 7 - 125/13
x = 76/13
Therefore, the solution to the system of equations x - 5y = 7 and 3x - 2y = -4 is x = 76/13 and y = -25/13.