1. Solve the system by any method:
2x-5y=-1
2x-y=1
How many solutions does the system of equations below have? Explain your answer.
y=3/5x-1
y=6/10x+3
plz help me ;-; i don't understand how to do this, sorry ;-;
1. 2x-y=1 => y=2x-1 ...
sub into equation 1 and solve for x ... 2x-5(2x-1)=-1 => 2x-10x+5=-1
=> 10x-2x=5+1
=> 8x=6 => x=(8/6)=(4/3)
Sub x into either equation and solve for y ...
2x-5y=-1
=> 2(4/3) - 5y = -1
=> 2.7 - 5y = -1
=> 2.7 + 1 = 5y
=> 3.7 = 5y
=> y = (3.7/5) = 0.74
Same procedure for 2nd set... Solve one equation for x or y, sub into the 2nd equation and solve for the variable.
Since both equations contain 2x, the easiest way would be elimination, thus subtract them:
-4y = -2
y = -2/-4 = 1/2
sub into the first:
2x - 5(1/2) = -1
2x = 5/2-1 = 3/2
x = 3/4 ---------> x = 3/4 , y = 1/2
#2
y=(3/5_x-1
y=6/10x+3 ----> y = (3/5)x + 3
since you have two lines with the same slope, but different y-intercepts, you have two distinct parallel lines.
Which of course cannot meet, so
there is no solution
No problem, I'm here to help! Let's start with the first system of equations:
1. Solve the system by any method:
2x - 5y = -1
2x - y = 1
To solve this system by any method, we can use the method of substitution or the method of elimination. Let's solve it using the method of substitution:
Step 1: Rearrange one of the equations to solve for one variable in terms of the other. Let's rearrange the second equation to solve for x in terms of y:
2x - y = 1
2x = y + 1
x = (y + 1)/2
Step 2: Substitute the expression for x in terms of y into the other equation. Let's substitute (y + 1)/2 for x in the first equation:
2((y + 1)/2) - 5y = -1
y + 1 - 5y = -1
-4y + 1 = -1
-4y = -2
y = -2/(-4)
y = 1/2
Step 3: Substitute the value of y back into one of the original equations to solve for x. Let's substitute y = 1/2 into the first equation:
2x - 5(1/2) = -1
2x - 5/2 = -1
2x = -1 + 5/2
2x = -2/2 + 5/2
2x = 3/2
x = (3/2)/2
x = 3/4
So, the solution to the first system of equations is x = 3/4 and y = 1/2.
Now let's move on to the second system of equations:
2. How many solutions does the system of equations below have? Explain your answer.
y = (3/5)x - 1
y = (6/10)x + 3
To determine the number of solutions, we can compare the slopes of the two equations. The slope-intercept form of a linear equation is y = mx + b, where m represents the slope.
In both equations, the slopes are (3/5) and (6/10), which simplify to 0.6 in both cases. Since the slopes are the same, the two lines are parallel and will never intersect. Therefore, the system of equations has no solution.
I hope this helps! Let me know if you have any further questions.
No problem! I'm here to help you. Let's start by solving the first system of equations:
1. Solve the system by any method:
2x - 5y = -1 ...(Equation 1)
2x - y = 1 ...(Equation 2)
We can solve this system of equations using the elimination method. To eliminate the x-term, we subtract Equation 2 from Equation 1:
(2x - 5y) - (2x - y) = (-1) - (1)
-5y + y = -2
-4y = -2
y = (-2) / (-4)
y = 1/2
Now, substitute the value of y into either Equation 1 or Equation 2 to find the value of x. Let's substitute it into Equation 2:
2x - (1/2) = 1
2x = 1 + (1/2)
2x = 3/2
x = (3/2) / 2
x = 3/4
So, the solution to the first system of equations is x = 3/4 and y = 1/2.
Moving on to the second question:
2. How many solutions does the system of equations below have? Explain your answer.
y = (3/5)x - 1 ...(Equation 1)
y = (6/10)x + 3 ...(Equation 2)
To determine the number of solutions, we need to compare the slopes of the two equations. The slope-intercept form of a line is y = mx + b, where m represents the slope.
In Equation 1, the slope is (3/5), and in Equation 2, the slope is (6/10) which simplifies to (3/5).
The fact that both equations have the same slope implies that the two lines are parallel. This means the system has no solution.
Therefore, the second system of equations has no solution.
I hope this explanation helps! Let me know if you have any further questions.