(a) Simplify
$(6x-3)(2x-1)$.
(b) Simplify
$(3x^2 - 2x^3 - 3x + 2)(x - 1)$.
(c) (i) Solve the simultaneous equations:
$2x - 3y = 18$
$5x + 9y = -10$.
(ii) V... show full transcript
Worked Solution & Example Answer:(a) Simplify
$(6x-3)(2x-1)$ - Junior Cycle Mathematics - Question Question 1
Step 1
Simplify $(6x-3)(2x-1)$
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Answer
To simplify the expression, we apply the distributive property:
(6x−3)(2x−1)=6x(2x)−6x(1)−3(2x)+3(1).
Calculating each term gives:
12x2
−6x
−6x
+3
Combining like terms results in:
12x2−12x+3.
Thus, the simplified form is:
Answer: 12x2−12x+3
Step 2
Simplify $(3x^2 - 2x^3 - 3x + 2)(x - 1)$
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Answer
First, distribute each term in (3x2−2x3−3x+2) by (x−1):
For 3x2, we get: 3x2imesx−3x2imes1=3x3−3x2,
For −2x3, we have:
−2x3imesx−(−2x3)imes1=−2x4+2x3,
For −3x, we find:
−3ximesx+3ximes1=−3x2+3x,
For 2, we can calculate:
2imesx−2imes1=2x−2.
Combining all these results:
−2x4+(3x3+2x3)−(3x2+3x2)+(2x+3x)−2.
Simplifying gives:
−2x4+5x3−6x2+5x−2.
Thus, the answer is:
Answer: −2x4+5x3−6x2+5x−2
Step 3
Solve the simultaneous equations: $2x - 3y = 18$ and $5x + 9y = -10$
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Answer
To solve these equations, we can use substitution or elimination. Here, we will use substitution:
Starting with the first equation:
2x−3y=18
Rearranging gives:
\Rightarrow x = \frac{3y + 18}{2}.$$
Now, substituting this into the second equation:
$$5\left( \frac{3y + 18}{2} \right) + 9y = -10.$$
Multiply everything by 2 to eliminate the fraction:
$$5(3y + 18) + 18y = -20 \
15y + 90 + 18y = -20 \
33y + 90 = -20 \
33y = -20 - 90 \
33y = -110 \
y = -\frac{110}{33} \
y = -\frac{10}{3}.$$
Substituting $y$ back to find $x$:
$$x = \frac{3(-\frac{10}{3}) + 18}{2}\
\Rightarrow x = \frac{-10 + 18}{2}\
\Rightarrow x = \frac{8}{2} \Rightarrow x = 4.$$
Thus, the solution to the simultaneous equations is:
**Answer: $x = 4$, $y = -\frac{10}{3}$**
Step 4
Verify your answer to (c)(i).
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Answer
To verify the solution x=4 and y=−310, using the first equation:
8 + 10 = 18.$$
This holds true. Now, using the second equation:
$$5(4) + 9(-\frac{10}{3}) = -10. \
20 - 30 = -10.$$
This also holds true. Thus, we have confirmed the answers are correct.
**Verification passed**
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