Determine whether the system of equations
has one solution, no solution, or infinitely
many solutions.
1. y - 13 = 5x
y - 5x = 12
2. y= 2x + 10
3 y - 6x = 30
4.y= 1/4x = -1
y - 2=4×
look at the equations in slope-intercept form:
y = 5x+13
y = 5x+12
Clearly, no value of x can make both of them the same.
Try this with the others. If the slopes are different, there will always be a solution, since the lines are not parallel.
To determine whether the systems of equations have one solution, no solution, or infinitely many solutions, we will solve each system of equations one by one:
1. y - 13 = 5x
y - 5x = 12
Let's solve this system of equations by the method of substitution:
From the first equation, we can rewrite it as y = 5x + 13.
Substituting this value of y in the second equation:
(5x + 13) - 5x = 12
5x - 5x + 13 = 12
13 = 12
This is not a true statement (13 ≠ 12), which means the system of equations has no solution.
2. y = 2x + 10
This is a linear equation in slope-intercept form. Since the equation has only one variable, it represents a straight line. Therefore, the system has infinitely many solutions.
3. y - 6x = 30
This is a linear equation with two variables and one equation, so it also represents a straight line. Therefore, the system has infinitely many solutions.
4. y = (1/4)x - 1
y - 2 = 4× (assuming a typo, we'll adjust to y - 2 = 4x)
Let's solve this system of equations by substitution:
Substitute the value of y from the first equation into the second equation:
(1/4)x - 1 - 2 = 4x
(1/4)x - 3 = 4x
(1/4)x - 4x = 3
(-15/4)x = 3
x = (3 * 4) / -15
x = -4/5
Substitute the value of x back into the first equation to find y:
y = (1/4)(-4/5) - 1
y = -1/5 - 1
y = -1/5 - 5/5
y = -6/5
So, the system of equations has one unique solution: x = -4/5, y = -6/5.
In summary:
1. No solution.
2. Infinitely many solutions.
3. Infinitely many solutions.
4. One solution with x = -4/5 and y = -6/5.
To determine whether the system of equations has one solution, no solution, or infinitely many solutions, we can do so by analyzing the equations and their coefficients.
1. y - 13 = 5x
y - 5x = 12
To solve this system, we can use the method of substitution.
First, let's isolate y in the first equation:
y = 5x + 13
Now, substitute this expression for y in the second equation:
(5x + 13) - 5x = 12
Simplifying this equation, we get:
13 = 12
The equation is not true. This means that there is no solution to this system of equations. Therefore, the system has no solution.
2. y = 2x + 10
This is a single equation in slope-intercept form, y = mx + b, where m is the coefficient of x and b is the y-intercept. This equation represents a straight line with a slope of 2 and a y-intercept of 10.
Since there is only one equation and one variable, this system has infinitely many solutions. Any value chosen for x will give us a corresponding value for y on this line.
3. y - 6x = 30
This equation represents a linear function. To determine whether it has one solution, no solution, or infinitely many solutions, we need more information. We would need another equation or condition to determine the relationship between y and x.
Without additional information, we cannot determine the number of solutions for this system.
4. y = 1/4x - 1
y - 2 = 4x
To solve this system, we can use the method of substitution.
First, substitute the expression for y in the second equation:
(1/4x - 1) - 2 = 4x
Simplifying this equation, we get:
1/4x - 3 = 4x
Rearranging the equation:
1/4x - 4x = 3
Combining the like terms, we get:
-15/4x = 3
To solve for x, we multiply both sides of the equation by -4/15:
x = -4/15 * 3
Simplifying, we get:
x = -4/5
Now, substitute this value of x back into one of the original equations (let's use the first one):
y = 1/4 * (-4/5) - 1
Simplifying, we get:
y = -4/20 - 1
y = -1/5 - 1
y = -1 - 1/5
y = -6/5
Therefore, the solution to this system of equations is x = -4/5 and y = -6/5. Thus, the system has one solution.
5. y - 2 = 4x
This equation is a linear function. Similar to the third example, without another equation or condition, we cannot determine the number of solutions for this system. Additional information is needed.