We are given two simultaneous linear equations: $x + y = 10$ and $y - 2x = 1$. We need to: a) Solve these equations for $x$ and $y$. b) Sketch the two equations on the same graph. c) Evaluate the point of intersection(s) in graph (b).

1. Problem Description

We are given two simultaneous linear equations: $x + y = 10$ and $y - 2x = 1$.

We need to:

a) Solve these equations for $x$ and $y$.

b) Sketch the two equations on the same graph.

c) Evaluate the point of intersection(s) in graph (b).

2. Solution Steps

a) Solving the equations:

We have the following system of equations:

$x + y = 10$ (Equation 1)

$y - 2x = 1$ (Equation 2)

From Equation 1, we can express $y$ in terms of $x$:

$y = 10 - x$ (Equation 3)

Substitute Equation 3 into Equation 2:

$(10 - x) - 2x = 1$

$10 - 3x = 1$

$-3x = 1 - 10$

$-3x = -9$

$x = \frac{-9}{-3}$

$x = 3$

Now, substitute the value of $x$ back into Equation 3 to find $y$:

$y = 10 - 3$

$y = 7$

Thus, the solution is $x = 3$ and $y = 7$.

b) Sketching the equations:

Equation 1: $x + y = 10$

When $x=0$, $y=10$. When $y=0$, $x=10$. So the line passes through $(0,10)$ and $(10,0)$.

Equation 2: $y - 2x = 1$ which can be written as $y = 2x + 1$.

When $x=0$, $y=1$. When $x=1$, $y=3$. So the line passes through $(0,1)$ and $(1,3)$.

Graphically sketching these lines would involve drawing a line through (0, 10) and (10, 0) and another line through (0, 1) and (1, 3).

c) Evaluating the point of intersection:

From part a), we found that the solution to the system of equations is $x=3$ and $y=7$. Therefore, the point of intersection of the two lines is $(3, 7)$. This point should lie on both sketched lines.

3. Final Answer

a) $x = 3$, $y = 7$

b) Sketch of $x+y=10$ passing through (0, 10) and (10, 0) and $y - 2x = 1$ passing through (0, 1) and (1, 3).

c) The point of intersection is $(3, 7)$.