Algebra - Introduction to Algebra

Simplify the expression: $4x + 7y - 2x + 3y + 5$

$2x + 10y + 5$

Group the like terms together: Group 1 ($x$ terms): $4x - 2x = 2x$ Group 2 ($y$ terms): $7y + 3y = 10y$ Group 3 (constants): $5$ Combine them to get $2x + 10y + 5$.

Expand and simplify: $3(2x - 4) + 2(x + 5)$

$8x - 2$

First, expand both brackets: $3 \times 2x = 6x$ $3 \times -4 = -12$ $2 \times x = 2x$ $2 \times 5 = 10$ This gives: $6x - 12 + 2x + 10$. Now, collect like terms: $6x + 2x = 8x$ $-12 + 10 = -2$ Result: $8x - 2$.

Factorize completely: $12x^2y - 18xy^2$

$6xy(2x - 3y)$

Identify the Highest Common Factor (HCF) for the coefficients and the variables: HCF of $12$ and $18$ is $6$. HCF of $x^2$ and $x$ is $x$. HCF of $y$ and $y^2$ is $y$. The total HCF is $6xy$. Divide each term by $6xy$ to find the terms inside the bracket: $\frac{12x^2y}{6xy} = 2x$ $\frac{18xy^2}{6xy} = 3y$ Result: $6xy(2x - 3y)$.

Evaluate the expression $2a^2 - 3b$ when $a = 3$ and $b = -4$.

$30$

Substitute the values into the expression: $2(3)^2 - 3(-4)$ Calculate the square first: $3^2 = 9$ $2(9) - 3(-4)$ Multiply: $18 - (-12)$ Subtracting a negative is the same as adding: $18 + 12 = 30$.

Solve for $x$: $5x - 7 = 13$

$x = 4$

To isolate $x$, first add $7$ to both sides: $5x = 13 + 7$ $5x = 20$ Next, divide both sides by $5$: $x = \frac{20}{5}$ $x = 4$.