Linear equations

The language of equations allows us to solve mysteries. Imagine you are playing "Guess my number" with a friend. Your friend tells you:

"If you add $12$ to my secret number, the result is $35$".

Translate this sentence into algebraic language, letting $x$ be the secret number, and write the corresponding equation.

In your favourite video game, you have achieved a score of $x$ points. After completing a daily quest, you were rewarded with $25$ extra points, and now you have a total of $110$ points.

Write the equation that describes this situation and work out in your head how many points you had at the beginning.

Checking whether a number is the solution to an equation is essential. State whether the value $x = 4$ is the solution to the following linear equations:

1. a) $x + 7 = 11$

1. b) $3x = 12$

1. c) $2x - 1 = 5$

Imagine you are preparing a test for your classmates and make up a linear equation that simultaneously satisfies these three conditions:

1. Its exact solution must be $x = -2$.

2. The unknown $x$ must necessarily appear on both sides of the equation.

3. At least one of the coefficients multiplying $x$ must be greater than $1$.

Clearly explain the reasoning you followed to construct your equation and prove that it is correct by checking that, when substituting $x = -2$, both sides give the same numerical value.

An equation is like a balance scale in perfect equilibrium. Look at the following image where the orange circles represent our unknown $x$ and the grey squares represent $1$ kg units.

Write the equation representing the balance of the scale:

Solve these very simple linear equations using inverse operations (if something adds, subtract it; if something multiplies, divide by it):

1. a) $x + 5 = 15$

1. b) $x - 9 = 2$

1. c) $4x = 24$

1. d) $\frac{x}{3} = 5$

Darragh tried to solve the equation $4x - 6 = 14$ in his notebook and wrote the following step-by-step resolution:

Step 1: $4x = 14 - 6$

Step 2: $4x = 8$

Step 3: $x = \dfrac{8}{4} = 2$

1. a) Identify the mistake Darragh made when going from the original statement to Step 1 and explain which basic rule of term transposition he did not follow.
2. b) Solve the equation correctly, showing all the steps.
3. c) Check your solution by substituting the value obtained into the initial equation to show that the equality holds.

A famous streamer is giving away gift cards to their subscribers. They announced that "three times the number of cards given away yesterday, minus the 12 saved for today, equals 48 cards".

If we let $x$ be the number of cards given away yesterday, write the equation that describes the situation and solve it to find that number.

To travel daily to high school and to his basketball practice, Seán analyses two fares for his Leap Card:

• Youth Fare (Young Adult Pass): a fixed monthly pass of $15$ euros plus $0.60$ euros per trip taken.

• Pay-as-you-go Fare (Leap Card Travel Credit): no fixed monthly fee, but each trip costs $1.10$ euros.

1. a) Set up a linear equation that allows calculating the number of trips $v$ for which the monthly cost would be identical in both options.
2. b) Solve the equation to find that number of trips.
3. c) If Seán estimates that he will take approximately $36$ trips per month, interpret the result obtained in the previous section to advise him on which of the two fares will be more economical for him. Justify your advice by comparing the expected costs.

Solve the following linear equation by first expanding the brackets and then grouping the terms with the unknown:

$4(x - 3) + 2x = 3(x + 2) + 9$

In a role-playing game, your character's strength is calculated using a "balance formula". To level up, you must find the value of the unknown $x$ that makes both sides of the magic balance scale equal. Solve the following equation:

$2(3x - 5) - 4(x - 2) = 16 - (x + 3)$

Find the value of $x$ that solves the following complex equation with nested brackets:

$3[2x - (x + 4)] = 2(3x + 1) - 5$

Make sure to solve the inner brackets first and simplify each side before rearranging terms.

Find a number such that half of it plus a quarter of it equals $15$. To solve it, find the lowest common multiple of the denominators to clear the equation of fractions.

$\frac{x}{2} + \frac{x}{4} = 15$

Solve the following equation with denominators, simplifying each step:

$\frac{2x - 1}{3} - \frac{x + 2}{6} = 2$

Remember to multiply all the terms in the equation by $m.c.m.(3, 6)$ to work with integers only.

Solve the following equation with denominators, paying close attention to the negative sign in front of the second fraction:

$\frac{x + 5}{2} - \frac{2x - 3}{3} = \frac{5}{6}$

Remember that to clear the denominators, you must multiply all terms by their LCM.

Given the equation with parentheses and fractions:

$\dfrac{3(x - 1)}{2} - \dfrac{2x + 4}{5} = 2$

Two students propose different solving strategies:

• Aoife's strategy: First multiply all the terms on both sides by the least common multiple of the denominators ($10$) to eliminate the fractions in one go, and then remove the parentheses.

• Cian's strategy: First apply the distributive property to the numerators, split each fraction into simple fractions, and operate by adding the terms with $x$ and the numerical terms using fraction arithmetic.

1. a) Solve the equation in an orderly manner following Aoife's strategy.
2. b) Solve the equation following Cian's strategy.
3. c) Compare both paths: which one do you consider more effective for avoiding common sign errors and simplifying intermediate calculations in 1st Year? Justify your answer by analysing a critical point where it is easy to make a mistake in either of the two methods.