Connect with knowledgeable experts and enthusiasts on IDNLearn.com. Our Q&A platform offers detailed and trustworthy answers to ensure you have the information you need.
Sagot :
Certainly! Let's solve this system of equations step-by-step using substitution and identify the solution.
Given system of equations:
[tex]\[ \begin{cases} y = -3x + 3 \\ 7x + 2y = 1 \end{cases} \][/tex]
### Step 1: Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]
The first equation already expresses [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ y = -3x + 3 \][/tex]
### Step 2: Substitute this expression into the second equation
Substitute [tex]\( y = -3x + 3 \)[/tex] into the second equation [tex]\( 7x + 2y = 1 \)[/tex]:
[tex]\[ 7x + 2(-3x + 3) = 1 \][/tex]
### Step 3: Simplify and solve for [tex]\( x \)[/tex]
Simplify the equation:
[tex]\[ 7x + 2(-3x) + 2(3) = 1 \][/tex]
[tex]\[ 7x - 6x + 6 = 1 \][/tex]
[tex]\[ x + 6 = 1 \][/tex]
Subtract 6 from both sides:
[tex]\[ x = 1 - 6 \][/tex]
[tex]\[ x = -5 \][/tex]
### Step 4: Substitute [tex]\( x = -5 \)[/tex] back into the first equation
To find [tex]\( y \)[/tex], substitute [tex]\( x = -5 \)[/tex] back into the first equation [tex]\( y = -3x + 3 \)[/tex]:
[tex]\[ y = -3(-5) + 3 \][/tex]
[tex]\[ y = 15 + 3 \][/tex]
[tex]\[ y = 18 \][/tex]
So, the solution to the system of equations is:
[tex]\[ (x, y) = (-5, 18) \][/tex]
### Step 5: Verify using the points given
The given points are:
[tex]\[ (18, -5), (2, 3), (4, -9), (-5, 18), (-9, 4), (3, 2) \][/tex]
We need to check which of these points are the solution to the system of equations:
[tex]\[ \begin{cases} y = -3x + 3 \\ 7x + 2y = 1 \end{cases} \][/tex]
The solution we found through substitution is [tex]\( (-5, 18) \)[/tex], which matches one of the given points. Therefore, the point that satisfies both equations is:
[tex]\[ (-5, 18) \][/tex]
Thus, the solution to the system of equations, verified against the given points, is:
[tex]\[ \boxed{(-5, 18)} \][/tex]
Given system of equations:
[tex]\[ \begin{cases} y = -3x + 3 \\ 7x + 2y = 1 \end{cases} \][/tex]
### Step 1: Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]
The first equation already expresses [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ y = -3x + 3 \][/tex]
### Step 2: Substitute this expression into the second equation
Substitute [tex]\( y = -3x + 3 \)[/tex] into the second equation [tex]\( 7x + 2y = 1 \)[/tex]:
[tex]\[ 7x + 2(-3x + 3) = 1 \][/tex]
### Step 3: Simplify and solve for [tex]\( x \)[/tex]
Simplify the equation:
[tex]\[ 7x + 2(-3x) + 2(3) = 1 \][/tex]
[tex]\[ 7x - 6x + 6 = 1 \][/tex]
[tex]\[ x + 6 = 1 \][/tex]
Subtract 6 from both sides:
[tex]\[ x = 1 - 6 \][/tex]
[tex]\[ x = -5 \][/tex]
### Step 4: Substitute [tex]\( x = -5 \)[/tex] back into the first equation
To find [tex]\( y \)[/tex], substitute [tex]\( x = -5 \)[/tex] back into the first equation [tex]\( y = -3x + 3 \)[/tex]:
[tex]\[ y = -3(-5) + 3 \][/tex]
[tex]\[ y = 15 + 3 \][/tex]
[tex]\[ y = 18 \][/tex]
So, the solution to the system of equations is:
[tex]\[ (x, y) = (-5, 18) \][/tex]
### Step 5: Verify using the points given
The given points are:
[tex]\[ (18, -5), (2, 3), (4, -9), (-5, 18), (-9, 4), (3, 2) \][/tex]
We need to check which of these points are the solution to the system of equations:
[tex]\[ \begin{cases} y = -3x + 3 \\ 7x + 2y = 1 \end{cases} \][/tex]
The solution we found through substitution is [tex]\( (-5, 18) \)[/tex], which matches one of the given points. Therefore, the point that satisfies both equations is:
[tex]\[ (-5, 18) \][/tex]
Thus, the solution to the system of equations, verified against the given points, is:
[tex]\[ \boxed{(-5, 18)} \][/tex]
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. IDNLearn.com is committed to providing the best answers. Thank you for visiting, and see you next time for more solutions.