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, 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.
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).