A. 3/5
B. −5/7
C. −1
D. 3
11 answers
To find the slope of the line passing through the points \((-3, 4)\) and \((4, -1)\), we can use the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \((x_1, y_1) = (-3, 4)\) and \((x_2, y_2) = (4, -1)\).
Substituting in the values:
\[ m = \frac{-1 - 4}{4 - (-3)} = \frac{-1 - 4}{4 + 3} = \frac{-5}{7} \]
So, the slope of the line is \(-\frac{5}{7}\).
The correct answer is:
B. \(-5/7\)
To determine which ordered pair is a solution to the equation \(2x + 3y = 16\), we will substitute each option into the equation and check if it holds true.
\[ 2(3) + 3(2) = 6 + 6 = 12 \quad \text{(Not a solution)} \]
\[ 2(11) + 3(1) = 22 + 3 = 25 \quad \text{(Not a solution)} \]
\[ 2(5) + 3(2) = 10 + 6 = 16 \quad \text{(This is a solution)} \]
\[ 2(7) + 3(2) = 14 + 6 = 20 \quad \text{(Not a solution)} \]
The ordered pair that is a solution to the equation \(2x + 3y = 16\) is:
C. (5, 2)
Let's analyze each statement for the given linear equation \(y = -2x + 4\).
A. The point (0, 2) lies on the graph of the equation.
To check this, substitute \(x = 0\):
\[ y = -2(0) + 4 = 4 \]
Since the result is \(4\) and not \(2\), this statement is false. The point (0, 2) does not lie on the graph.
B. The graph of the equation is the set of all points on the coordinate plane.
This statement is false. The graph of the equation is a straight line, not the entire plane.
C. The point (1, 2) lies on the graph of the equation.
To check this, substitute \(x = 1\):
\[ y = -2(1) + 4 = -2 + 4 = 2 \]
Since the result is \(2\), the point (1, 2) does lie on the graph of the equation. This statement is true.
D. The graph of the equation is the set of points that are solutions to the equation.
This statement is true. The graph of a linear equation represents all the points \((x, y)\) that satisfy the equation, which are indeed solutions.
Summary of True Statements:
To convert the equation \(y + 2 = 13x\) into standard form, we need to rearrange it into the form \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) should be non-negative.
Starting with the given equation:
\[ y + 2 = 13x \]
First, subtract \(2\) from both sides:
\[ y = 13x - 2 \]
Now, rearrange this to isolate \(y\) on one side:
\[ -13x + y = -2 \]
To write it in standard form, we want to move \(13x\) to the right side:
\[ 13x + y = -2 \]
Since \(A\) (which is \(13\)) is already positive, we can leave it as is.
Thus, the standard form of the equation is:
\[ 13x + y = -2 \]
To convert the equation \(y + 2 = \frac{1}{3}x\) into standard form \(Ax + By = C\), we need to rearrange it while ensuring \(A\), \(B\), and \(C\) are integers.
Starting with the given equation:
\[ y + 2 = \frac{1}{3}x \]
First, subtract \(2\) from both sides:
\[ y = \frac{1}{3}x - 2 \]
Next, to eliminate the fraction, multiply all terms by \(3\) (the denominator):
\[ 3y = x - 6 \]
Now, rearranging this into standard form, we want to have all terms on one side:
\[ -x + 3y = -6 \]
If we prefer \(A\) to be positive, we can multiply the entire equation by \(-1\):
\[ x - 3y = 6 \]
Now, we can write it in standard form:
\[ x - 3y = 6 \]
Thus, the standard form of the equation is:
\[ x - 3y = 6 \]
To convert the equation \(y + 2 = 13x\) to standard form \(Ax + By = C\), follow these steps:
Start with the equation: \[ y + 2 = 13x \]
Subtract \(2\) from both sides: \[ y = 13x - 2 \]
Rearrange to move \(13x\) to the left side: \[ -13x + y = -2 \]
To express it in the desired format \(Ax + By = C\), we can multiply the entire equation by \(-1\) to make the coefficient of \(x\) positive: \[ 13x - y = 2 \]
Now it's in standard form: \[ 13x - y = 2 \]
So, filling in the blanks: 13 + (-1) = 2.
