Question
Solve the simultaneous equations.
$$\begin{array}{l}5t-2w=19\\3t+2w=5\end{array}$$
▶️Answer/Explanation
$[t = ] 3$
$[w = ] –2$
Since the coefficients of \( w \) are +2 and -2
adding the equations will eliminate \( w \).
$
(5t – 2w) + (3t + 2w) = 19 + 5
$
$
5t + 3t = 24
$
$
8t = 24
$
$
t = 3
$
Substitute \( t = 3 \) into one equation
$
3(3) + 2w = 5
$
$
9 + 2w = 5
$
$
2w = 5 – 9
$
$
2w = -4
$
$
w = -2
$
$
t = 3, \quad w = -2
$
Question
Solve the simultaneous equations. You must show all your working.
$4y+3x=13$
$y=x^2-18$
▶️Answer/Explanation
$4x^{2}+3x-85[=0]$
or 16$y^2-113y+7[=0]$
oe simplified
correct method to solve their quadratic equation e.g. factors, quadratic formula, completing the square
$$\begin{aligned}&x=-5\:y=7\\&x=\frac{17}{4}\:\mathrm{oe}\:y=\frac{1}{16}\:\mathrm{oe}\end{aligned}$$
1. \( 4y + 3x = 13 \)
2. \( y = x^2 – 18 \)
substitute \( y = x^2 – 18 \) into the first equation
$
4(x^2 – 18) + 3x = 13
$
$
4x^2 – 72 + 3x = 13
$
$
4x^2 + 3x – 72 – 13 = 0
$
$
4x^2 + 3x – 85 = 0
$
using the quadratic formula
$
x = \frac{-3 \pm \sqrt{3^2 – 4(4)(-85)}}{2(4)}
$
$
= \frac{-3 \pm \sqrt{9 + 1360}}{8}
$
$
= \frac{-3 \pm \sqrt{1369}}{8}
$
Since \( \sqrt{1369} = 37 \):
$
x = \frac{-3 \pm 37}{8}
$
1. \( x = \frac{-3 + 37}{8} = \frac{34}{8} = 4.25 \)
2. \( x = \frac{-3 – 37}{8} = \frac{-40}{8} = -5 \)
For \( x = 4.25 \):
$
y = (4.25)^2 – 18
$
$
= 0.0625
$
For \( x = -5 \):
$
y = (-5)^2 – 18
$
$
= 7
$
$
(x, y) = (4.25, 0.0625) \quad \text{or} \quad (-5, 7)
$
Question
Solve the simultaneous equations. You must show all your working.
$$\frac{3x}{2}+5y=5\\4x-3y=46$$
▶️Answer/Explanation
Correctly equating one set of coefficients
Correct method to eliminate one variable
$x = 10, y = –2$
$
\frac{3x}{2} + 5y = 5 \quad \text{(Equation 1)}
$
$
4x – 3y = 46 \quad \text{(Equation 2)}
$
$
2 \times \left( \frac{3x}{2} + 5y = 5 \right)
$
$
3x + 10y = 10 \quad \text{(Equation 3)}
$
From Equation 3
$
3x = 10 – 10y
$
$
x = \frac{10 – 10y}{3}
$
Substitute this value of \( x \) into Equation 2
$
4x – 3y = 46
$
$
4\left( \frac{10 – 10y}{3} \right) – 3y = 46
$
$
\frac{4(10) – 4(10y)}{3} – 3y = 46
$
$
40 – 40y – 9y = 138
$
$
40 – 49y = 138
$
$
-49y = 98
$
$
y = -2
$
Substitute \( y = -2 \)
$
x = \frac{10 – 10(-2)}{3}
$
$
x = \frac{10 + 20}{3}
$
$
x = 10
$
Question
Solve the inequality \(\frac{2x-3}{5}-\frac{x}{3}\leq 2\)
Answer/Explanation
Ans: x ≤ 39 www
Question
Solve the simultaneous equations.
3x + 5y = 24
x + 7y = 56
Answer/Explanation
Ans: x = −7
y = 9
Question
Solve the simultaneous equations.
You must show all your working.
5x + 2y = –2
3x – 5y = 17.4
Answer/Explanation
Ans: Correctly equating one set of coefficients
Correct method to eliminate one variable
x = 0.8
y = −3