Equation 1 must pass through (3,- 5) and (- 1, 7), while Equation 2 only passes through (- 1, 7).
y=- 3x+4 y=2x+9
Practice makes perfect
This exercise asks us to write a system of linear equations that satisfies the given conditions. To begin, let's make sense of the conditions.
Condition #1: Equation 1 must pass through the point (3,- 5).
Condition #2: Equation 2 must not pass through the point (3,- 5).
Condition #3: The solution to the system is (- 1,7).
By considering these conditions, let's write Equation 1 and Equation 2 one at a time.
Equation 1
To satisfy Conditions #1 and #3, Equation 1 must pass through the points ( 3, - 5) and ( - 1, 7). We can use these points to determine the slope of the line using the Slope Formula.
Slope=y_2-y_1/x_2-x_1 ⇔ 7-( -5)/( -1)- 3= - 3
Thus, the slope of Equation 1 is - 3. Now we will write Equation 1 in point-slope form using the slope and either of the points. Let's use ( 3, - 5).
To satisfy Conditions #2 and #3, Equation 2 must pass through (- 1, 7) and not pass through (3, - 5). To ensure this is true we will first use the point ( - 1, 7) in point-slope form.
y-y_1 &= m(x-x_1)
&⇕
y- 7 &= m(x-( - 1))
As we have ensured that both lines pass through (- 1, 7), the solution to the system will be (- 1, 7). To make sure that the system only has one solution, the slopes of the lines must be different. Once we determine the slope of Equation 1, we will choose any different value for the slope of Equation 2. Let's arbitrarily choose to use m= 2 for the slope.
