Review the graphs of a system of two linear equations in two
variables: x - 5y = 7 and
3х - 2y = -4. Find the solution to both equations.
(1 point)
The intersection point is
Explain
To review the graphs of the system of linear equations, let's start by rearranging both equations to slope-intercept form (y = mx + b):
1) x - 5y = 7
-5y = -x + 7
y = (1/5)x - 7/5
2) 3x - 2y = -4
-2y = -3x - 4
y = (3/2)x + 2
Now, we can see that the slopes of the two lines are different. The slope of the first equation (1/5) is smaller than the slope of the second equation (3/2). Therefore, the lines will intersect at a single point.
To find the intersection point, we can set the two equations equal to each other:
(1/5)x - 7/5 = (3/2)x + 2
Let's solve for x:
Multiply through by 10 to get rid of the fractions:
2x - 14 = 15x + 20
Combine like terms:
2x - 15x = 20 + 14
-13x = 34
Divide by -13:
x = -34/13
Now substitute this value of x back into either equation to solve for y. Let's use the first equation:
y = (1/5)(-34/13) - 7/5
Simplify:
y = -34/65 - 91/65
y = -125/65
Therefore, the solution to the system of equations is x = -34/13 and y = -125/65 (-2.62, -1.92 rounded to two decimal places).
The intersection point is where the two lines cross, and in this case, it is approximately (-2.62, -1.92).
To review the graphs of the system of equations, let's start by finding the slope-intercept form of each equation.
The first equation: x - 5y = 7
To write it in slope-intercept form, we isolate y:
-5y = -x + 7
Divide by -5:
y = (1/5)x - 7/5
The second equation: 3x - 2y = -4
Similarly, isolate y:
-2y = -3x - 4
Divide by -2:
y = (3/2)x + 2
Now, let's plot the graphs of these equations on a coordinate plane.
The first equation, y = (1/5)x - 7/5, has a y-intercept of -7/5 and a slope of 1/5. This means that for every increase of 5 in x, y increases by 1. So we can plot two points, connect them, and draw the line.
The second equation, y = (3/2)x + 2, has a y-intercept of 2 and a slope of 3/2. Similarly, for every increase of 2 in x, y increases by 3. We can plot two points, connect them, and draw the line.
After plotting both lines, we can observe that they intersect at a single point. This point is the solution to both equations. It represents the values of x and y that satisfy both equations simultaneously.
To find the exact solution, we can equate the two equations:
(1/5)x - 7/5 = (3/2)x + 2
Simplifying the equation, we get:
(1/5)x - (3/2)x = 2 + 7/5
(-13/10)x = 17/5
To solve for x, we can multiply both sides by -10/13:
x = -(170/13)
Substitute this value back into either of the original equations to find y.
For example, let's use the first equation:
(170/13) - 5y = 7
Simplify:
-5y = 7 - (170/13)
-5y = (91/13)
Divide both sides by -5:
y = -(91/65)
Therefore, the solution to both equations is x = -(170/13) and y = -(91/65). This is the intersection point of the two lines on the graph.
To find the solution to the system of equations, we can graph each equation and find the intersection point.
Let's start with the first equation: x - 5y = 7.
To graph this equation, we can rewrite it in slope-intercept form: y = (1/5)x - 7/5. This equation represents a line with a slope of 1/5 and a y-intercept of -7/5.
Now, let's graph the second equation: 3x - 2y = -4.
Similarly, we can rewrite this equation in slope-intercept form: y = (3/2)x + 2. This equation represents a line with a slope of 3/2 and a y-intercept of 2.
Now we have both equations graphed on the same coordinate plane. The intersection point of the two lines represents the solution to the system of equations.
To find the coordinates of the intersection point, we can visually identify the point where the two lines intersect on the graph.
Once you have determined the coordinates of the intersection point, that point represents the solution to the system of equations.
Note: If the lines are parallel and do not intersect, it means that the system of equations has no solution. If the lines overlap and are the same, it means that the system of equations has infinitely many solutions.