Answer:
[tex]\displaystyle a=\frac{7}{2}\text{ or } 3.5[/tex]
Step-by-step explanation:
We have the two points (3a, 4) and (a, -3).
And we want to find the value of a such that the gradient of the line joining the two points is 1.
Recall that the gradient or slope of a line is given by the formula:
[tex]\displaystyle m=\frac{y_2-y_1}{x_2-x_1}[/tex]
Where (x₁, y₁) is one point and (x₂, y₂) is the other.
Let (3a, 4) be (x₁, y₁) and (a, -3) be (x₂, y₂). Substitute:
[tex]\displaystyle m=\frac{-3-4}{a-3a}[/tex]
Simplify:
[tex]\displaystyle m=\frac{-7}{-2a}=\frac{7}{2a}[/tex]
We want to gradient to be one. Therefore, m = 1:
[tex]\displaystyle 1=\frac{7}{2a}[/tex]
Solve for a. Rewrite:
[tex]\displaystyle \frac{1}{1}=\frac{7}{2a}[/tex]
Cross-multiply:
[tex]2a=7[/tex]
Therefore:
[tex]\displaystyle a=\frac{7}{2}\text{ or } 3.5[/tex]
Answer:
[tex] \frac{7}{2} [/tex]
Step-by-step explanation:
Objective: Linear Equations and Advanced Thinking.
If a line connects two points (3a,4) and (a,-3) has a gradient of 1. This means that the slope formula has to be equal to 1
If we use the points to find the slope: we get
[tex] \frac{4 + 3}{3a - a} [/tex]
Notice how the numerator is 7, this means the denominator has to be 7. This means the denomiator must be 7.
[tex]3a - a = 7[/tex]
[tex]2a = 7[/tex]
[tex]a = \frac{7}{2} [/tex]
