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 (
).
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To find the intersection point of the two equations, we can solve the system of equations using any method such as substitution or elimination.
Let's solve the system of equations using the substitution method:
First, isolate one of the variables in one equation. We'll isolate x in the first equation:
x - 5y = 7
x = 7 + 5y
Now substitute this expression for x into the second equation:
3(7 + 5y) - 2y = -4
21 + 15y - 2y = -4
21 + 13y = -4
13y = -4 - 21
13y = -25
y = -25/13 ≈ -1.923
Now substitute this value of y back into the expression for x:
x = 7 + 5(-25/13) = 7 - (125/13) = (91 - 125)/13 = -34/13 ≈ -2.615
So the solution to the system of equations is (x, y) = (-34/13, -25/13) or approximately (-2.615, -1.923).
To graph the system of equations, we can plot the two lines and find their intersection point:
To graph the equation x - 5y = 7, we can rewrite it in slope-intercept form:
-5y = -x + 7
y = (1/5)x - 7/5
To graph the equation 3x - 2y = -4, we can rewrite it in slope-intercept form:
-2y = -3x - 4
y = (3/2)x + 2
Now we can plot the two lines on a coordinate plane:
The line for the equation y = (1/5)x - 7/5 has a y-intercept of -7/5 and a slope of 1/5, meaning that for every increase of 1 in x, y increases by 1/5.
The line for the equation y = (3/2)x + 2 has a y-intercept of 2 and a slope of 3/2, meaning that for every increase of 1 in x, y increases by 3/2.
The two lines intersect at the point (-34/13, -25/13).
Overall, the graphs of the system of equations and the solution to both equations are visually represented by the intersection point (-34/13, -25/13).
To find the solution to both equations, we can start by graphing them on a coordinate plane.
The first equation is x − 5y = 7:
To graph this equation, we can start by finding the x-intercept and y-intercept.
For the x-intercept, we set y = 0:
x − 5(0) = 7
x = 7
So, the x-intercept is (7, 0).
For the y-intercept, we set x = 0:
0 − 5y = 7
-5y = 7
y = -7/5
So, the y-intercept is (0, -7/5).
Plotting these points, we can draw a line passing through both points:
Next, let's graph the second equation: 3x − 2y = −4.
Again, we can start by finding the x-intercept and y-intercept.
For the x-intercept, we set y = 0:
3x − 2(0) = −4
3x = −4
x = -4/3
So, the x-intercept is (-4/3, 0).
For the y-intercept, we set x = 0:
3(0) − 2y = −4
-2y = -4
y = 2
So, the y-intercept is (0, 2).
Plotting these points, we can draw a line passing through both points:
Now, let's find the solution to both equations.
The solution to a system of linear equations is the point (x, y) that satisfies both equations.
To find the solution, we can solve the system of equations by substitution or elimination.
Substituting x from the first equation into the second equation:
3(7 - 5y) - 2y = -4
21 - 15y - 2y = -4
21 - 17y = -4
-17y = -25
y = 25/17
Substituting this value of y back into the first equation:
x - 5(25/17) = 7
17x - 125/17 = 119/17
17x = 119/17 + 125/17
17x = 119/17 + 125/17
17x = 244/17
x = 244/17
So, the solution to the system of equations is (244/17, 25/17).
Therefore, the intersection point is (244/17, 25/17).
To find the solution to a system of two linear equations in two variables, you can use the method of graphing.
Step 1: Graph the first equation, x - 5y = 7.
Rearrange the equation to get it in the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
x - 5y = 7 becomes -5y = -x + 7, and then y = (1/5)x - 7/5.
This equation represents a line with a slope of 1/5 and a y-intercept of -7/5.
To graph it, plot the y-intercept at (0, -7/5) and then use the slope to find more points on the line. For example, if you go up 1 unit and to the right 5 units, you get the point (5, -6/5).
Connect the points to draw the line.
Step 2: Graph the second equation, 3x - 2y = -4.
Rearrange the equation to get it in the slope-intercept form.
3x - 2y = -4 becomes -2y = -3x - 4, and then y = (3/2)x + 2.
This equation represents a line with a slope of 3/2 and a y-intercept of 2.
Plot the y-intercept at (0, 2) and use the slope to find more points on the line. For example, if you go up 3 units and to the right 2 units, you get the point (2, 7).
Draw the line.
Step 3: Find the intersection point of the two lines.
The solution to the system of equations is the point where the two lines intersect.
Looking at the graph, you can see that the lines intersect at the point (2, 1).
Therefore, the solution to the system of equations x - 5y = 7 and 3x - 2y = -4 is (2, 1).