Test: Simultaneous (Linear) Equations (Including Problems) - Year 12 MCQ

The general approach to finding ages based on ratios is to set up equations using defined variables for each person's age. This systematic method allows for accurate solutions based on the relationships provided in the problem.

The relationship between cost price \( x \) and selling price \( y \) can be expressed as \( y = x + (profit \% \times x) \). This shows that the selling price is the cost price plus the profit made on it.

The cross-multiplication method utilizes a specific formula to find the values of \( x \) and \( y \) directly from two linear equations. By identifying the coefficients and applying the formula, one can solve for both variables efficiently.

The first step in solving age-related problems presented in a ratio is to define variables for the ages involved. This allows for the formulation of equations based on the relationships described.

Adding the equations gives \( 2x = 8 \) or \( x = 4 \). Substituting \( x = 4 \) into \( x + y = 7 \) gives \( y = 3 \). Therefore, the solution is \( x = 4, y = 3 \).

To eliminate \( y \), one effective method is to substitute \( y \) from the first equation \( y = 10 - x \) into the second equation. This will simplify the problem into a single variable equation.

The first step in using the elimination method for these equations is to multiply one or both equations to make the coefficients of either \( x \) or \( y \) equal, allowing for elimination when the equations are added or subtracted.

To express \( y \) in terms of \( x \) from the equation \( x + y = 7 \), you rearrange it to get \( y = 7 - x \). This expression is crucial for methods like substitution.

To find two numbers based on their sum and difference, you set up two equations: one for their sum and another for their difference. Solving this system provides the values of both numbers effectively.

In simultaneous equations, the "solution" refers to the specific pair of values for the variables that satisfy all equations in the system. This means that these values make all the equations true simultaneously.

The primary method for solving equations that have variables in the denominators involves variable substitution, such as letting \( 1/x = a \) or \( 1/y = b \). This transformation simplifies the problem into a linear format, making it easier to resolve.

Simultaneous linear equations are commonly used in real-life situations such as budgeting, where one might need to determine how to allocate a fixed amount of money across multiple expenses, like tickets and snacks.

In the method of elimination by equating coefficients, one or both equations are multiplied by necessary factors to make the coefficients of one variable equal. This allows for elimination through addition or subtraction of the equations, leading to a simpler equation to solve.

The solution \( x = 2, y = 1 \) represents the point of intersection of the two lines defined by the simultaneous equations. This point signifies the values of \( x \) and \( y \) that satisfy both equations simultaneously.

If the product of two fractions equals 1, it indicates that they are reciprocals of each other. For example, \( \frac{a}{b} \times \frac{b}{a} = 1 \).

The correct representation of A's age \( x \) and B's age \( y \) in a ratio of 9:4 is \( 9x - 4y = 0 \). This equation can be used in conjunction with another equation to find the actual ages.

Let the two numbers be \( x \) and \( y \). The equations formed are \( x + y = 12 \) and \( x - y = 2 \). Adding these gives \( 2x = 14 \) or \( x = 7 \). Substituting \( x \) back into the first equation gives \( y = 5 \). Thus, the numbers are 7 and 5.

Cross-multiplication is particularly effective for solving equations with variables in denominators. It allows for a straightforward approach to eliminate the fractions, transforming the equation into a more manageable form.

The standard form of a linear equation is \( ax + by + c = 0 \), where \( a \), \( b \), and \( c \) are real numbers, and \( x \) and \( y \) are variables. This format is essential as it allows for easy identification of the coefficients and constant, which aids in graphing and solving the equation.

The first step in the substitution method is to express one variable in terms of the other. This allows for substitution into the second equation, simplifying the system to a single-variable equation that can be easily solved.