What is an equation of a line perpendicular to the line with equation
y = 1 - 3x
a.y=-3x + 5 b. y=3x + 5
c.y=-1/3x + 5 d. y=1/3x + 5
To solve the linear system below, which substitution of unkowns is proper ?
x + 4y= -6
5x - 3y = -16
a. substitute -4y-6 for x in the first equation
b. substitute 4y - 6 for x in the second equation
c.substitute 3y + 16 for x in the second equation
d. -4y - 6 for x in the second equation
What is the solution of the equation
3(1/4x - 1/4)= 4x - 1 ?
Using the formula for interest, I = Prt, solve for r.
please solve these problems clearly so I can follow and understand them, thank you .
I'll check your answers.
To find the equation of a line perpendicular to the line y = 1 - 3x, we need to determine the slope of the given line first. The slope-intercept form of a line is y = mx + b, where m is the slope.
Comparing the given equation y = 1 - 3x with the slope-intercept form, we see that the slope is -3.
The equation of a line perpendicular to a given line will have a slope that is the negative reciprocal of the slope of the given line.
The negative reciprocal of -3 is 1/3.
So, the equation of the line perpendicular to y = 1 - 3x will have a slope of 1/3.
Now, let's look at the answer choices:
a. y = -3x + 5: This equation does not represent a line with a slope of 1/3.
b. y = 3x + 5: This equation does not represent a line perpendicular to y = 1 - 3x.
c. y = (-1/3)x + 5: This equation represents a line with a slope of -1/3, not 1/3.
d. y = (1/3)x + 5: This equation represents a line with a slope of 1/3, which is perpendicular to y = 1 - 3x.
Therefore, the correct answer is d. y = (1/3)x + 5.
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To solve the linear system of equations x + 4y = -6 and 5x - 3y = -16, we can use substitution.
Let's look at the answer choices:
a. substitute -4y - 6 for x in the first equation: This substitution is not correct because it gives us an expression (-4y - 6) rather than a value for x.
b. substitute 4y - 6 for x in the second equation: This substitution is not correct because it gives us an expression (4y - 6) rather than a value for x.
c. substitute 3y + 16 for x in the second equation: This substitution is not correct because it gives us an expression (3y + 16) rather than a value for x.
d. substitute -4y - 6 for x in the second equation: This substitution is correct because it gives us a specific value (-4y - 6) for x.
Now, let's substitute -4y - 6 for x in the second equation:
5(-4y - 6) - 3y = -16
-20y - 30 - 3y = -16
-23y - 30 = -16
-23y = 14
y = -14/23
Substitute the value of y back into the first equation:
x + 4(-14/23) = -6
x - 56/23 = -6
x = -6 + 56/23
x = (-138 + 56)/23
x = -82/23
Therefore, the solution to the linear system of equations is x = -82/23 and y = -14/23.
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To solve the equation 3(1/4x - 1/4) = 4x - 1, we can simplify the equation step by step.
First, distribute the 3 to the terms inside the parentheses:
3/4x - 3/4 = 4x - 1
Next, let's get rid of the fractions by multiplying both sides of the equation by 4 to get rid of the denominators:
4(3/4x - 3/4) = 4(4x - 1)
3x - 3 = 16x - 4
Now, isolate the terms with x on one side of the equation:
3x - 16x = -4 + 3
-13x = -1
Finally, divide both sides of the equation by -13 to solve for x:
x = -1 / -13
x = 1/13
Therefore, the solution to the equation is x = 1/13.
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To solve the formula for interest, I = Prt, for r (the interest rate), we need to isolate "r" on one side of the equation.
The formula for interest, I = Prt, can be rearranged as follows:
I = Prt
Divide both sides by Pt:
I / Pt = r
Therefore, the formula for r is r = I / Pt.
1. Equation of a line perpendicular to y = 1 - 3x:
To find the equation of a line that is perpendicular, we need to find the negative reciprocal of the slope of the given line. The given line has a slope of -3, so the negative reciprocal is 1/3.
Thus, the equation of a line perpendicular to y = 1 - 3x is in the form y = (1/3)x + b, where b is the y-intercept.
Among the given options, the equation that matches this form is d. y = (1/3)x + 5.
2. Proper substitution of variables for the linear system:
To solve the linear system x + 4y = -6 and 5x - 3y = -16, we need to eliminate one of the variables by substituting an equivalent expression from one equation into the other. The goal is to create an equation in one variable.
To determine the proper substitution, we should isolate one variable in one equation and substitute it into the other equation.
In this case, option b. substitute 4y - 6 for x in the second equation is the correct choice. We can isolate x in the first equation and substitute it into the second equation so that we have one equation with only y.
The first equation: x = -4y - 6
Substituting x = -4y - 6 into the second equation: 5(-4y - 6) - 3y = -16
This simplifies to -17y - 30 = -16, which leads to -17y = 14, and finally, y = -14/17.
3. Solution of the equation 3(1/4x - 1/4) = 4x - 1:
To solve this equation, we distribute the 3 to the terms inside the parentheses and then simplify the equation step by step.
Expanding the left side: 3 * 1/4x - 3 * 1/4 = 4x - 1
This simplifies to 3/4x - 3/4 = 4x - 1
Now, we can get rid of the fractions by multiplying every term in the equation by the common denominator, which is 4.
Multiplying both sides by 4: 4(3/4x - 3/4) = 4(4x - 1)
This simplifies to 3x - 3 = 16x - 4
Next, we gather like terms on each side of the equation: 3x - 16x = -4 + 3
This simplifies to -13x = -1, and dividing both sides by -13 gives x = 1/13.
4. Solving for r in the formula I = Prt:
The formula I = Prt represents the equation for interest, where I is the interest, P is the principal amount, r is the interest rate, and t is the time in years.
To solve for r, we need to isolate r on one side of the equation.
Dividing both sides of the equation by Pt, we have: I / Pt = r
Thus, the formula for r is r = I / Pt.