Question
Fill in the blanks so that the following system of linear equations has infinite solutions. Click Here for Help Video
𝑦
=
−
8
𝑥
+
8
y=−8x+8
𝑦
=
y=
𝑥
+
x+
𝑦
=
−
8
𝑥
+
8
y=−8x+8
𝑦
=
y=
𝑥
+
x+
Answers
GPT-4o mini
To have infinite solutions for the system of linear equations, both equations must represent the same line. This means that the second equation must be equivalent to the first equation, which is given as:
\[ y = -8x + 8 \]
To express this line in a different form, we can rearrange or manipulate the equation. The second equation is given in the form:
\[ y = x + b \]
To determine \( b \), we need the slopes of both equations to match, which means we need to convert the first equation into the form of the second equation.
1. The slope of the first equation, \( y = -8x + 8 \), is \(-8\).
2. For the second equation \( y = x + b \), the slope is \(1\).
To have the same slope, we need to manipulate the first equation accordingly. Instead of using an additive constant, we can express it in another way.
A simple way to find a matching second line is to multiply the first equation by \(-\frac{1}{8}\):
If we multiply by \(-\frac{1}{8}\):
\[ y = -\frac{1}{8}(-8x + 8) \]
This gives:
\[ y = x - 1 \]
Notice that this does not represent infinite solutions. Instead, we maintain the same constants for infinite solutions.
Let's try to modify \(b\):
Setting the equation for infinite solutions:
We must have:
1. The \(x\)-term from the first equation is matched to the second equation
2. The coefficients must be the same or proportionally equatable.
Thus, if rewriting \(b\) to complete it, we have:
- Second equation in proper form as \(y = -8x + b\) with \(b = 8\)
Then the complete equations will be:
1. \(y = -8x + 8\)
2. \(y = -8x + 8\) or expressed as \(y = x + 8\)
Thus, the filled version of your equation must be:
\[ y = -8x + 8 \]
\[ y = -8x + 8 \]
In conclusion, the answer for your specific solution has:
**Blank 1**: \(-8\)
**Blank 2**: \(8\)
\[ y = -8x + 8 \]
To express this line in a different form, we can rearrange or manipulate the equation. The second equation is given in the form:
\[ y = x + b \]
To determine \( b \), we need the slopes of both equations to match, which means we need to convert the first equation into the form of the second equation.
1. The slope of the first equation, \( y = -8x + 8 \), is \(-8\).
2. For the second equation \( y = x + b \), the slope is \(1\).
To have the same slope, we need to manipulate the first equation accordingly. Instead of using an additive constant, we can express it in another way.
A simple way to find a matching second line is to multiply the first equation by \(-\frac{1}{8}\):
If we multiply by \(-\frac{1}{8}\):
\[ y = -\frac{1}{8}(-8x + 8) \]
This gives:
\[ y = x - 1 \]
Notice that this does not represent infinite solutions. Instead, we maintain the same constants for infinite solutions.
Let's try to modify \(b\):
Setting the equation for infinite solutions:
We must have:
1. The \(x\)-term from the first equation is matched to the second equation
2. The coefficients must be the same or proportionally equatable.
Thus, if rewriting \(b\) to complete it, we have:
- Second equation in proper form as \(y = -8x + b\) with \(b = 8\)
Then the complete equations will be:
1. \(y = -8x + 8\)
2. \(y = -8x + 8\) or expressed as \(y = x + 8\)
Thus, the filled version of your equation must be:
\[ y = -8x + 8 \]
\[ y = -8x + 8 \]
In conclusion, the answer for your specific solution has:
**Blank 1**: \(-8\)
**Blank 2**: \(8\)