First-degree equations
You already know first-degree equations from earlier years. Here is a quick revision, focusing on setting them up from a problem: that is the part most often asked for in 4th ESO.
The balance: revision
Solve the following linear equation by first expanding the brackets: $4(x - 3) - 2(x + 1) = 5(x - 2) + 4$
Solve the following linear equation by first expanding the brackets:
$4(x - 3) - 2(x + 1) = 5(x - 2) + 4$
Brackets and denominators
The order that always works: remove brackets → remove denominators (multiplying by the LCM) → collect the $x$ terms on one side → solve for $x$ → check.
Calculate the value of $x$ in the following equation with denominators: $\frac{2x - 1}{3} + \frac{x + 2}{4} = 5$
Calculate the value of $x$ in the following equation with denominators:
$\frac{2x - 1}{3} + \frac{x + 2}{4} = 5$
Solve the following linear equation involving fractions and nested brackets: $\frac{3(x - 2)}{4} - \frac{2(x + 1)}{6} = \frac{x}{3} - \frac{1}{2}$
Solve the following linear equation involving fractions and nested brackets:
$\frac{3(x - 2)}{4} - \frac{2(x + 1)}{6} = \frac{x}{3} - \frac{1}{2}$
Find the solution to the equation by working with decimals: $0,5(2x - 4) + 1,2(x + 3) = 3,4x - 0,8$
Find the solution to the equation by working with decimals:
$0,5(2x - 4) + 1,2(x + 3) = 3,4x - 0,8$
Setting up a problem with an equation
The most important step is not solving: it is translating the statement into an equation. Name the unknown $x$ and write each piece of data as an operation with $x$.
The perimeter of a rectangular plot is $48\text{ m}$. Knowing that the base is exactly twice the height, set up an equation to find the lengths of the sides of the plot.
The perimeter of a rectangular plot is $48\text{ m}$. Knowing that the base is exactly twice the height, set up an equation to find the lengths of the sides of the plot.
In a coffee shop, two types of coffee beans are mixed. The first costs $6\text{ \euro/kg}$ and the second $8\text{ \euro/kg}$. If $50\text{ kg}$ of mixture is required…
In a coffee shop, two types of coffee beans are mixed. The first costs $6\text{ \euro/kg}$ and the second $8\text{ \euro/kg}$. If $50\text{ kg}$ of mixture is required at a final price of $6,60\text{ \euro/kg}$, set up an equation to find out how many kilograms of each type of coffee should be used.
Two freight trains leave two stations A and B separated by a distance of $450\text{ km}$ at the same time, travelling on parallel tracks in opposite directions. The…
Two freight trains leave two stations A and B separated by a distance of $450\text{ km}$ at the same time, travelling on parallel tracks in opposite directions. The train leaving A travels at $120\text{ km/h}$ and the one leaving B at $80\text{ km/h}$. Set up an equation to determine how long it will take for them to pass each other and at what distance from station A they will meet.
Rearranging formulae
The same rules can be used to isolate one letter in a formula containing several, which is very useful in physics and economics.
In a multiple-choice exam consisting of $40\text{ questions}$, $3\text{ points}$ is awarded for each correct answer, $1\text{ point}$ is deducted for each incorrect…
In a multiple-choice exam consisting of $40\text{ questions}$, $3\text{ points}$ is awarded for each correct answer, $1\text{ point}$ is deducted for each incorrect answer, and unanswered questions neither add nor subtract points. A student left $4\text{ questions}$ unanswered and obtained a final score of $60\text{ points}$. Set up a linear equation to calculate how many questions they answered correctly.
An electronics shop applies a discount of $15\%$ to the retail price of a laptop. At the checkout, $21\%$ VAT is added to the already discounted price. If the customer…
An electronics shop applies a discount of $15\%$ to the retail price of a laptop. At the checkout, $21\%$ VAT is added to the already discounted price. If the customer pays $822,80\text{ \euro}$ in the end, set up a linear equation to calculate the initial price of the laptop before the discount and before VAT.