Question
Which shows the correct first step to solving the system of equations in the most efficient manner?
3 x + 2 y = 17. x + 4 y = 19.
x = negative 4 y + 19
x = StartFraction negative 2 y + 17 Over 3 EndFraction
4 y = negative x + 19
y = StartFraction negative 3 x + 17 Over 2 EndFraction
3 x + 2 y = 17. x + 4 y = 19.
x = negative 4 y + 19
x = StartFraction negative 2 y + 17 Over 3 EndFraction
4 y = negative x + 19
y = StartFraction negative 3 x + 17 Over 2 EndFraction
Answers
GPT-4o mini
To solve the given system of equations:
1. \(3x + 2y = 17\)
2. \(x + 4y = 19\)
Let’s identify the most efficient first step.
One common method to solve systems of equations is substitution or elimination. In this case, let's examine the provided equations closely and see what we can derive from them:
- The second equation can be rearranged for \(x\):
\[
x = 19 - 4y
\]
Alternatively, we could express y in terms of x from the first equation or find another useful transformation from the combinations of provided options.
Here are the options given:
1. \(x = -4y + 19\)
2. \(x = \frac{-2y + 17}{3}\)
3. \(4y = -x + 19\)
4. \(y = \frac{-3x + 17}{2}\)
Let's check if these expressions correctly relate \(x\) and \(y\) from the original equations:
1. **Option 1**: \(x = -4y + 19\)
- This is a valid rearrangement of the second equation, \(x + 4y = 19\), so this option is correct.
2. **Option 2**: \(x = \frac{-2y + 17}{3}\)
- Rearranging the first equation would not yield this form, so this option does not relate correctly.
3. **Option 3**: \(4y = -x + 19\)
- This can be transformed into \(x + 4y = 19\), so this option is also a correct representation of the second equation but does not isolate either variable.
4. **Option 4**: \(y = \frac{-3x + 17}{2}\)
- This is a rearrangement of the first equation \(3x + 2y = 17\), which can be derived.
Given the efficiency of steps and clear expression of one variable in terms of the other, the best first step appears to be:
**Option 1: \(x = -4y + 19\)**
This choice allows for straightforward substitution into the other equation, leading to solving the system efficiently.
1. \(3x + 2y = 17\)
2. \(x + 4y = 19\)
Let’s identify the most efficient first step.
One common method to solve systems of equations is substitution or elimination. In this case, let's examine the provided equations closely and see what we can derive from them:
- The second equation can be rearranged for \(x\):
\[
x = 19 - 4y
\]
Alternatively, we could express y in terms of x from the first equation or find another useful transformation from the combinations of provided options.
Here are the options given:
1. \(x = -4y + 19\)
2. \(x = \frac{-2y + 17}{3}\)
3. \(4y = -x + 19\)
4. \(y = \frac{-3x + 17}{2}\)
Let's check if these expressions correctly relate \(x\) and \(y\) from the original equations:
1. **Option 1**: \(x = -4y + 19\)
- This is a valid rearrangement of the second equation, \(x + 4y = 19\), so this option is correct.
2. **Option 2**: \(x = \frac{-2y + 17}{3}\)
- Rearranging the first equation would not yield this form, so this option does not relate correctly.
3. **Option 3**: \(4y = -x + 19\)
- This can be transformed into \(x + 4y = 19\), so this option is also a correct representation of the second equation but does not isolate either variable.
4. **Option 4**: \(y = \frac{-3x + 17}{2}\)
- This is a rearrangement of the first equation \(3x + 2y = 17\), which can be derived.
Given the efficiency of steps and clear expression of one variable in terms of the other, the best first step appears to be:
**Option 1: \(x = -4y + 19\)**
This choice allows for straightforward substitution into the other equation, leading to solving the system efficiently.