Answer :
To determine how many solutions exist for the given system of equations:
1. [tex]\( y = 7x - 3 \)[/tex]
2. [tex]\(-14x + 2y = -3\)[/tex]
We will analyze the relationship between these two equations to see if they intersect at any point, which would represent a solution to the system.
Step 1: Rewrite the equations, if necessary, to easily compare them.
The first equation is already in slope-intercept form:
[tex]\[ y = 7x - 3 \][/tex]
The second equation is:
[tex]\[ -14x + 2y = -3 \][/tex]
Step 2: Substitute the expression for [tex]\( y \)[/tex] from the first equation into the second equation.
Substituting [tex]\( y = 7x - 3 \)[/tex] into [tex]\(-14x + 2y = -3\)[/tex]:
[tex]\[ -14x + 2(7x - 3) = -3 \][/tex]
Step 3: Simplify the substituted equation.
Distribute the 2 inside the parentheses:
[tex]\[ -14x + 14x - 6 = -3 \][/tex]
Combine the [tex]\( x \)[/tex] terms:
[tex]\[ 0x - 6 = -3 \][/tex]
[tex]\[ -6 = -3 \][/tex]
Step 4: Determine the validity of the resulting equation.
The equation [tex]\(-6 = -3\)[/tex] is a contradiction. This statement is not true and indicates that there is no value of [tex]\( x \)[/tex] that can satisfy both original equations simultaneously.
Conclusion:
Since the substitution resulted in a contradiction, there are no solutions to the system of equations. Therefore, the system of equations has:
0 solutions.
1. [tex]\( y = 7x - 3 \)[/tex]
2. [tex]\(-14x + 2y = -3\)[/tex]
We will analyze the relationship between these two equations to see if they intersect at any point, which would represent a solution to the system.
Step 1: Rewrite the equations, if necessary, to easily compare them.
The first equation is already in slope-intercept form:
[tex]\[ y = 7x - 3 \][/tex]
The second equation is:
[tex]\[ -14x + 2y = -3 \][/tex]
Step 2: Substitute the expression for [tex]\( y \)[/tex] from the first equation into the second equation.
Substituting [tex]\( y = 7x - 3 \)[/tex] into [tex]\(-14x + 2y = -3\)[/tex]:
[tex]\[ -14x + 2(7x - 3) = -3 \][/tex]
Step 3: Simplify the substituted equation.
Distribute the 2 inside the parentheses:
[tex]\[ -14x + 14x - 6 = -3 \][/tex]
Combine the [tex]\( x \)[/tex] terms:
[tex]\[ 0x - 6 = -3 \][/tex]
[tex]\[ -6 = -3 \][/tex]
Step 4: Determine the validity of the resulting equation.
The equation [tex]\(-6 = -3\)[/tex] is a contradiction. This statement is not true and indicates that there is no value of [tex]\( x \)[/tex] that can satisfy both original equations simultaneously.
Conclusion:
Since the substitution resulted in a contradiction, there are no solutions to the system of equations. Therefore, the system of equations has:
0 solutions.
