Verified step-by-step solutions for your class notes.
Verify if the ordered pair $(4, 3)$ is a solution of $5x - 6y = 2$. Explain your reasoning.
Answer: Substitute $x = 4$ and $y = 3$ into the equation $5x - 6y = 2$.
LHS = $5(4) - 6(3) = 20 - 18 = 2$.
RHS = 2.
Since LHS = RHS, the ordered pair $(4, 3)$ is a solution of the equation.
Find any two solutions for each of the following equations:
(i) $7x - 3y = 21$
(ii) $2x + 3y = 5$
Answer:
(i) $7x - 3y = 21$
Let $x = 0$: $7(0) - 3y = 21 \implies -3y = 21 \implies y = -7$. Solution 1: $(0, -7)$.
Let $y = 0$: $7x - 3(0) = 21 \implies 7x = 21 \implies x = 3$. Solution 2: $(3, 0)$.
(ii) $2x + 3y = 5$
Let $x = 1$: $2(1) + 3y = 5 \implies 3y = 3 \implies y = 1$. Solution 1: $(1, 1)$.
Let $y = 3$: $2x + 3(3) = 5 \implies 2x = -4 \implies x = -2$. Solution 2: $(-2, 3)$.
(Other valid solutions exist).
In the equations given below, m and n are unknown constants:
$2mx + 3y = 7$
$4x + ny = -10$
If $(2, 1)$ is the solution of both equations, find the values of m and n.
Answer: Substitute $x = 2$ and $y = 1$ into both equations.
First equation: $2m(2) + 3(1) = 7 \implies 4m + 3 = 7 \implies 4m = 4 \implies \mathbf{m = 1}$.
Second equation: $4(2) + n(1) = -10 \implies 8 + n = -10 \implies \mathbf{n = -18}$.
Consider the graph of the equation $3x - 7y = 21$ shown below. Does the point C $(2, 3)$ lie on the line? Does it satisfy the equation? Can points that do not lie on the line satisfy the equation?
Answer:
- Look at the graph: Point C $(2, 3)$ is clearly floating above the green line, so it does not lie on the line.
- Let's check algebraically: Substitute $x=2, y=3$ into $3x - 7y = 21$.
LHS = $3(2) - 7(3) = 6 - 21 = -15$. Since $-15 \neq 21$, it does not satisfy the equation.
- No, points that do not lie on the line can never satisfy the equation, because the line is the exact graphical representation of all the solutions to that equation.
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